Modal Logic With Non-deterministic Semantics: Part II -- Quantified Case
Marcelo E. Coniglio, Luis Fariñas del Cerro, Newton M. Peron
aa r X i v : . [ m a t h . L O ] J a n Modal Logic With Non-deterministic Semantics:Part II - Quantified Case
Marcelo E. Coniglio , Luis Fari˜nas del Cerro and Newton M. Peron Institute of Philosophy and the Humanities (IFCH) andCentre for Logic, Epistemology and The History of Science (CLE),University of Campinas (UNICAMP), Campinas, SP, Brazil.E-mail: [email protected] IRIT, Universit´e de Toulouse, CNRS, France.E-mail: [email protected] Graduate Program in Philosophy,Federal University of Southern Frontier (UFFS), Chapec´o, SC, Brazil.E-mail: newton.peron@uffs.edu.br
Abstract
In the first part of this paper we analyzed finite non-deterministic matrix semantics for propositionalnon-normal modal logics as an alternative to the standard Kripke’s possible world semantics. Thiskind of modal systems characterized by finite non-deterministic matrices was originally proposed byJu. Ivlev in the 70’s. The aim of this second paper is to introduce a formal non-deterministic semanticalframework for the quantified versions of some Ivlev-like non-normal modal logics. It will be shown thatseveral well-known controversial issues of quantified modal logics, relative to the identity predicate,Barcan’s formulas, and de dicto and de re modalities, can be tackled from a new angle within thepresent framework.
Keywords:
First-order modal logic; non-normal modal logic; non-deterministic matrices;Ivlev’s logics; Barcan formulas; de re and de dicto modalities; contingent identities.
Introduction
In previous papers (see [10, 11, 12]), we analyzed finite non-deterministic matrix semantics forpropositional modal logics as an alternative to the standard Kripke’s possible world semantics,in order to better understand the modal concepts of ‘necessary’ and ‘possible’ at the proposi-tional level. This kind of semantics was independently proposed by Y. Ivlev (see [21, 22, 23, 24])and J. Kearns (see [25]). The aim of this paper, which is the second part of [12], is to introducea formal non-deterministic semantical framework for some non-normal first-order modal logics,based on Ivlev’s approach to propositional non-normal modal logics. In such works, Ivlev in-troduced several modal systems which do not have the necessitation rule, for instance, weakerversions of T and S5 . The semantics proposed by Ivlev is given by finite-valued Nmatricestogether with the notion of (single-valued) valuations considered by A. Avron and I. Lev in [1] Closely related results were independently obtained by Omori and Skurt in [32]. (which also introduced the terminology non-deterministic matrices/Nmatrices and legal valu-ations ). Thus, Ivlev’s systems constitute one of the earliest antecedents of Avron and Lev’snon-deterministic semantics. The quantified version of the Ivlev-like modal logics presentedhere is along the same lines as the non-deterministic semantics proposed in [13] for quantifiedparaconsistent logics. As we shall see along this paper, several well-known criticisms to quan-tified modal logics, relative to the identity predicate, Barcan’s formulas, and de dicto and dere modalities, can be tackled from a new angle within the present framework.In Part I of this paper ([12]), Ivlev’s propositional systems were expanded and it was shown,among other results, that several Ivlev-like modal systems, which are characterized by finite-valued non-deterministic semantics, can be captured in terms of the modal concepts of nec-essarily true , possibly true and actually true . In particular, the four-valued systems can becaptured by only two concepts: actually true and contingently true .Now, let us consider the quantified case. In first-order classical logic, a (usually called
Tarskian ) structure is a pair A = h U, · A i such that U is a nonempty set called the domain or universe of the structure, in which the individuals of the structure exist. The function · A assigns a concrete interpretation for the symbols of the signature. In particular, n -ary predicatesymbols are interpreted as set of n -tuples over U , which is the extension of the predicate in A . In Kripke semantics for first-order modal logic, there are at least two universes for thatstructure. First, there is a set of worlds and a relation between them. Besides this, thefunction A associates to each predicate its extension in a given world. In a constant domain approach, there is a fixed nonempty set U which states the individuals that exist in every world.In a varying domain approach, the set U is replaced by a function U ( w ) that stablishes thedomain of the individuals that exist in the world w . Let Tm be the Ivlev’s four-valued version of T , and let us denote by Tm ∗ the proposedquantified version of it. Since the four-valued non-deterministic semantics for Tm can bedescribed in terms of the concepts of actually true and contingently true , this suggest that thepredicate symbols in a first-order structure A for Tm ∗ could be interpreted in terms of twomappings a A and c A , describing the actual and the contingent extension of P in A . In moregeneral systems involving six or eight truth-values, which are explained in terms of the modalconcepts of necessarily true , possibly true and actually true , each predicate could be interpretedin terms of three mappings describing the respective extensions.The organization of this paper is as follows: Section 1 presents the semantical intuitionsbehind the non-deterministic framework for the modal systems proposed here. In Section 2 thebasic four-valued system Tm ∗ is introduced, as a possible first-order version of Ivlev’s four-valued system Tm . As we shall see, at least two possible (non-deterministic) interpretations forthe quantifiers could be considered. Section 3 analyzes some well-known philosophical questionsrelated to the standard approach to quantified modal logic, showing that some of the criticismsto quantified modal logics can be can be avoided or better controlled in Tm ∗ . In particular,it will be shown that the choice of one or another interpretation for the quantifiers in Tm ∗ is related to the distinction between de dicto and de re modalities. In Section 4, first-orderversions of other four-valued, six-valued and eight-valued Ivlev-like modal systems are brieflydiscussed. Finally, Section 5 discusses the results presented along the paper as well as somefuture lines of research. Other antecedents of Nmatrices were proposed by N. Rescher [37], Quine [34], Kearns [25], and J. Crawfordand D. Etherington [15]. This is closely related to the swap-structures semantics proposed in [14] for some of these systems. See, for instance, [30, p. 49] and [16, p. 80–81]. For constant domains, see [20, p. 243] and [17, p. 95-98]. For varying domains, see [20, p. 278–279] and[17, p. 101–104].
The semantical intuitions behind (first-order) non-deterministic modal semantics According to the brief discussion presented in the Introduction, the four-valued non-deterministicsemantics for Tm can be described in terms of the concepts of actually true and contingentlytrue , hence the predicate symbols in a structure A for Tm ∗ can be interpreted in terms oftwo mappings a A and c A . To this end, in this paper we will consider semantical structuresfor a non-deterministic four-valued first-order modal logic as being pairs A = h U, · A i such thatthe interpretation under · A of function symbols and individual constants is defined as usualin Tarskian first-order structures. On the other hand, each n -ary predicate symbol P will beinterpreted as a pair P A ( P ) = ( a A ( P ) , c A ( P )) of subsets of U n . The function a A says whenevera tuple of individuals of the domain actually satisfies or not predicate P . In other words, thefunction a A assigns to the predicate P its actual extension in A . In turn, the function c A as-signs to P its contingent extension in A . It says whenever a tuple of individuals of the domain contingently satisfies or not predicate P . These two modal concepts produce four modal valuesthat can be interpreted as follows: T + : necessarily true; C + : contingently true; C − : contingently false; F − : necessarily false / impossible.Thus, the interpretation of a given n -ary predicate P in a four-valued structure A as abovegives origin to the following configuration over U n : a A ( P ) c A ( P ) T + C + C − F − The truth-value attached to each of the four areas above is the value assigned to the atomicformula P ( τ , . . . , τ n ) when the n -tuple ( τ A , . . . , τ A n ) of U n associated to ( τ , . . . , τ n ) in A belongsto that area. For instance, P ( τ , . . . , τ n ) gets the value T + in A iff ( τ A , . . . , τ A n ) ∈ a A ( P ) \ c A ( P ).In Remark 2.6 it will be shown that this approach is equivalent to consider, for every n -ary predicate symbol P , a function P A : U n → V , where V is the set of four truth-valuescorresponding to the four areas in the figure above. In order to better understand the first-order expansion of this semantics in a intuitive way,let us consider the following sentences as example:(1) Socrates is mortal This idea was inspired on the first-order extension of Kleene’s three-valued logic K3 proposed by Kripkein [28]. Observe that the interpretation of predicate symbols as mappings from the domain of a first-order structureinto the domain of an algebra corresponds to what is usually done in the realm of algebraic semantics for first-order logics.
The semantical intuitions behind (first-order) non-deterministic modal semantics The sentence (1) is necessarily true when the individual Socrates is in the actual extensionof the predicate “mortal” but not in the contingent extension of it. But (1) is only contingentlytrue when Socrates is both in the actual and in the contingent extension of the predicate“mortal”. If Socrates is not in the actual but is in the contingent extension of the predicate“mortal”, we say that (1) is contingently false. Finally, if Socrates is neither in the actual norin the contingent extension of the predicate “mortal”, we conclude that (1) is impossible.Consider now the universal sentence:(2) All are mortalsThe sentence (2) will be necessarily true when it is necessarily true for each individual ofthe domain. In this case, the actual extension of the predicate “mortal” must coincide withthe domain. Besides, the contingent extension of “mortal” must be empty. The sentence (2) isonly contingently true if: (i) there is at least one individual of the domain that has contingentlythe propriety of being mortal; and (ii) each individual of the domain is necessarily mortal oronly contingently mortal. In order to guarantee both conditions, we require that the actualextension of “mortal” coincides with the domain, but its contingent extension must not beempty.We say that (2) is contingently false if at least one individual of the domain has not thepropriety of being mortal. Besides, any individual of the domain could be mortal, that is, theyhave contingently the propriety of being mortal. The idea here is that the actual extension of“mortal” does not coincide with the domain but the union of the actual and the contingentextension of “mortal” does. Finally, (2) is impossible when it is not possible to at least oneindividual to have the propriety of being mortal, that is, at least one thing is not actually neithercontingently mortal. We can state that condition saying that there is at least one individual ofthe domain that is not in the union of the actual and the contingent extension of “mortal”.With respect to equality, consider the sentence:(3) Phosphorus is HesperusIn Kripke semantics (with constant or varying domains), proper names refer to the sameindividual through all the possible worlds. Because of this, they are called rigid designators . Iftwo proper names refer to the same individual of the domain in some possible world, they willrefer to the same one through all the possible worlds, that is, equalities are always necessary. Conversely — with certain restrictions on the accessibility relation between the worlds in ourKripkean models — if two proper names refer to different individuals in some possible world,they will refer to different ones through all the possible worlds. That means, inequalities arenecessary too. Those considerations force us to admit that, in Kripke semantics, (3) is either necessarilytrue or necessary false, that is, impossible. In this section, we will present an analogous approachconsidering first-order version of Ivlev semantic, that is, (3) will receive only two values: thevalue “necessarily true” when “Phosphorus” and “Hesperus” refer to the same individual inthe domain; the value “impossible” if they don’t. However, this can be seen as an arbitraryoption. In Subsection 3.1, we will discuss how Ivlev’s semantics can also deal with contingentidentities in a natural way. See [29, p. 48]. The sentence (3) is an example given by Kripke in [29, p. 28–29]. In [20, p. 311], the authors showed also that in Kripkean models in which the relation between possibleworld is, at least, reflexive and symmetrical, the following holds:( x y ) → (cid:3) ( x y ) . This proves that in some Kripkean systems not only equalities of any kind (between constants or functions) arenecessary but also that inequalities of any kind are necessary.
The first-order non-normal modal logic Tm ∗ Tm ∗ In this section the intuitive ideas described in Section 1 will be formalized. Recall that Tm is one of the (non-normal) propositional modal systems introduced by Ivlev in [23], but alsostudied independently by Kearns in [25], and more recently by Omori and Skurt in [32] andby us in [10, 11, 12]. An outstading feature of Tm is that it is semantically characterized bya four-valued non-deterministic matrix (or Nmatrix), by using the terminology and formalismintroduced by Avron and Lev in [1] (see also [2]). Thus, Ivlev’s modal logics constitute an earlyantecendent of Nmatrices (called quasi matrices by Ivlev in [23]). Tm and two ways of defining quantifiers over it The propositional modal logic Tm is defined over the propositional signature Σ, which con-sists of the connectives ¬ (negation), (cid:3) (necessary) and → (implication). Semantically, Tm is characterized by a four-valued non-deterministic matrix M Tm = hA Tm , D i such that A Tm = h V , · Tm i is a multialgebra over Σ (recall [12, Definition 2.1]) with domain V = { T + , C + , C − , F − } and D = { T + , C + } is the set of designated truth-values. From now on the multioperation Tm associated to each connective A Tm as follows: x ˜ ¬ xT + { F − } C + { C − } C − { C + } F − { T + } x ˜ (cid:3) xT + { T + , C + } C + { C − , F − } C − { C − , F − } F − { C − , F − } ˜ → T + C + C − F − T + { T + } { C + } { C − } { F − } C + { T + } { T + , C + } { C − } { C − } C − { T + } { T + , C + } { T + , C + } { C + } F − { T + } { T + } { T + } { T + } Disjunction and conjunction can be defined in Tm as follows: α ∨ β := ¬ α → β and α ∧ β := ¬ ( α → ¬ β ), while possibility is given as usual by ♦ α := ¬ (cid:3) ¬ α . The correspondingmultioperators are defined in A Tm as follows: x ˜ ♦ xT + { T + , C + } C + { T + , C + } C − { T + , C + } F − { C − , F − } ˜ ∨ T + C + C − F − T + { T + } { T + } { T + } { T + } C + { T + } { T + , C + } { T + , C + } { C + } C − { T + } { T + , C + } { C − } { C − } F − { T + } { C + } { C − } { F − } ˜ ∧ T + C + C − F − T + { T + } { C + } { C − } { F − } C + { C + } { C + } { F − , C − } { F − } C − { C − } { F − , C − } { F − , C − } { F − } F − { F − } { F − } { F − } { F − } It is easy to see that x ˜ ∨ y = ˜ ¬ ( ˜ ¬ x ˜ ∧ ˜ ¬ y ) and x ˜ ∧ y = ˜ ¬ ( ˜ ¬ x ˜ ∨ ˜ ¬ y ) for every x, y ∈ V . Thatis, the De Morgan rules are valid in A Tm .In Subsection 2.2 a first-order extension of Tm called Tm ∗ will be proposed. The se-mantics of Tm ∗ will be given by means of first-order structures A evaluated over the Nmatrix The first-order extension of other Ivlev-like modal systems as the ones studied in the first part of thispaper [12] can be done in an analogous way, by means of straightforward adaptations.
The first-order non-normal modal logic Tm ∗ M Tm . This approach to first-order logics with a non-deterministic semantics based on Nmatri-ces was already considered in the literature for several paraconsistent logics known as logics offormal inconsistency , see [3] and [13]. The latter considers a family of Nmatrices defined overcertain non-deterministic algebras called swap structures . These structures, which generalizethe finite-valued Nmatrices, were developed in [14] for several Ivlev-like modal logics. As weshall see, the semantics for Tm ∗ adapted from [3] and [13] will be equivalent to the semanticalapproach informally described in the previous section.The first step is extending the Nmatrix M Tm with a multioperator ˜ Q : ( P ( V ) − {∅} ) → ( P ( V ) − {∅} ) for every quantifier Q ∈ {∀ , ∃} . The idea is that a given valuation over the ex-tended Nmatrix will choose a value, for a given formula of the form Qxϕ , within the set ˜ Q ( X ),where X is the set of instances of ϕ ( x ) over the given first-order structure A . Accordinglyto the previous approaches to quantified Nmatrices, which are inspired in algebraic quantifiedlogics, it is natural that ˜ ∀ ( X ) and ˜ ∃ ( X ) be defined as the conjunction and the disjunctionof the members of X according to the respective multioperators of A Tm . This produces thefollowing: X ˜ ∀ ( X ) { T + } { T + }{ C + } { C + }{ T + , C + } { C + }{ C − , C + } { F − , C − }{ C − , C + , T + } { F − , C − }{ C − } { C − }{ C − , T + } { C − } F − ∈ X { F − } X ˜ ∃ ( X ) T + ∈ X { T + }{ C + } { C + }{ C + , F − } { C + }{ C + , C − } { T + , C + }{ C + , C − , F − } { T + , C + }{ C − } { C − }{ C − , F − } { C − }{ F − } { F − } It is easy to see that ˜ ∀ ( X ) = ˜ ¬ ˜ ∃ ( ˜ ¬ X ) and ˜ ∃ ( X ) = ˜ ¬ ˜ ∀ ( ˜ ¬ X ), where, for every ∅ 6 = X ⊆ V , ˜ ¬ X = { ˜ ¬ x : x ∈ X } . However, since we are interested in the satisfaction of theBarcan formulas (see Subsection 3.4), and taking into consideration the semantic intuitions ofthe sentence (2) explored in section 1, stricter (deterministic) forms of quantification will beconsidered in Tm ∗ , namely ˜ ∀ d and ˜ ∃ d , given by the tables above. X ˜ ∀ d ( X ) { T + } { T + }{ C + } { C + }{ T + , C + } { C + }{ C − , C + } { C − }{ C − , C + , T + } { C − }{ C − } { C − }{ C − , T + } { C − } F − ∈ X { F − } X ˜ ∃ d ( X ) T + ∈ X { T + }{ C + } { C + }{ C + , F − } { C + }{ C + , C − } { C + }{ C + , C − , F − } { C + }{ C − } { C − }{ C − , F − } { C − }{ F − } { F − } It is easy to see that the deterministic quantifiers correspond, respectively, to the deter-ministic conjunction and disjunction of the members of X according to the order given by thechain F − ≤ C − ≤ C + ≤ T + . Clearly ˜ ∀ d ( X ) = ¬ ˜ ∃ d ( ¬ X ) and ˜ ∃ d ( X ) = ¬ ˜ ∀ d ( ¬ X ) for every ∅ 6 = X ⊆ V .As we shall see in Subsection 3.3, in order to analyze the distinction between the de re and de dicto modalities, both kinds of quantifiers ˜ Q and ˜ Q d will be relevant. For ‘instances of ϕ ( x ) over A ’ we mean the set of denotations of ϕ over the structure A , when x takes allthe possible values in the domain of A , assuming that any variable occurring free in ϕ other than x has beninterpreted by a given assignment over A . The technical details will be given in Subsection 2.3. The first-order non-normal modal logic Tm ∗ Tm ∗ and its axiomatics To our purposes, we will consider first-order modal languages based on the propositional signa-ture Σ described at the beginning of Subsection 2.1, expanded with the universal quantifier ∀ , and defined over first-order signatures, which are defined as usual. Thus, a first-order signature is a collection Θ formed by the following symbols: (i) a non-empty set of predicate symbols P , with the corresponding arity ̺ ( P ) ≥ P ∈ P ; (ii) a possible empty set F offunction symbols, with the corresponding arity ̺ ( f ) ≥ f ∈ F ; (iii) a possible emptyset of individual constants C . It will also assumed a fixed denumerable set V ar = { x , x , . . . } of individual variables. Give a first-order signature Θ, the set
T er (Θ) of terms over Θ is defined recursively asfollows: (i) a variable or a constant is a term; (ii) if f is a n -ary function and τ , . . . , τ n areterms, then f τ . . . τ n is a term. The set F or (Θ) of well-formed formulas (wffs) is also definedrecursively as follows: (i) for each n -ary predicate P , if τ , . . . , τ n are terms, then P τ . . . τ n isa wff (called atomic); in particular, if τ and τ are terms then ≈ τ τ , which will be written as( τ ≈ τ ), is an atomic wff; (ii) if α is a wff and x is a variable, then ( ¬ α ), ( (cid:3) α ) and ( ∀ xα ) arealso wffs; (iii) if α and β are wffs, then ( α → β ) is also a wff; (iv) nothing else is a wff. We willeliminate parenthesis when the readability is unambiguous.The following abbreviations will be used in Tm ∗ : α ∨ β := ¬ α → β (disjunction); α ∧ β := ¬ ( α → ¬ β ) (conjunction); ♦ α := ¬ (cid:3) ¬ α (possibility); ∃ xα := ¬∀ x ¬ α (existential quantifier).Let α be a wff of a first-order modal language F or (Θ) over a first-order signature Θ. Thenotion of free and bounded occurrences of a variable in a formula, as well as the notion of termfree for a variable in a formula, closed term (that is, without variables) and closed formula (orsentence, that is, a formula without free occurrences of variables) are defined as usual (see,for instance, [30]). The set of closed formulas and closed terms over Θ will be denoted by
Sen (Θ) and
CT er (Θ), respectively. We write α [ x/τ ] to denote the formula obtained from α by replacing simultaneously every free occurrence of the variable x by the term τ . If α is aformula and y is a variable free for the variable x in α , α [ x ≀ y ] denotes any formula obtainedfrom α by replacing some, but not necessarily all (maybe none), free occurrences of x by y . Definition 2.1 (da Costa) . Let ϕ and ψ be formulas. If ϕ can be obtained from ψ by meansof addition or deletion of void quantifiers, or by renaming bound variables (keeping the samefree variables in the same places), we say that ϕ and ψ are variant of each other. Definition 2.2 (Hilbert calculus for Tm ∗ ) . The first-order modal logic Tm ∗ is defined by aHilbert calculus which consists of the following axiom schemas and inference rules: As discussed in the previous subsection, conjunction ∧ , disjunction ∨ and existential quantifier ∃ can bedefined in terms of the other symbols, hence it will be ommited from the list of primitive symbols. Indeed, it will be assumed a symbol ≈ for the identity predicate such that ̺ ( ≈ ) = 2. Most part of modal logic manuals — for instance [20], [17] and [19] — do not use function symbols amongthe symbols of their first-order modal signature (an exception is [8, p. 241]). However, it is standard the useof function symbols in first-order signatures for classical logic, see for instance [30, p. 49] and [16, p. 70]. Itseems that this absence in approaches to first-oder modal logic is related to the problem of contingent identities,as we will discuss in Subsection 3.1. We choose a wider approach by including function symbols, since in thisnew semantic the treatment of contingent identities is radically different comparing to Kripkean approach. Asa result, some problems concerning references through the possible worlds will simply disappear. That is, a quantifier ∀ xα such that x does not occur free in α . Recalling that ♦ α is an abbreviation for ¬ (cid:3) ¬ α . The first-order non-normal modal logic Tm ∗ (Ax1) α → ( β → α ) (Ax2) ( α → ( β → γ )) → (( α → β ) → ( α → γ )) (Ax3) ( ¬ β → ¬ α ) → (( ¬ β → α ) → β ) (Ax4) ∀ xα → α [ x/τ ] if τ is free for x in α (Ax5) ∀ x ( α → β ) → ( α → ∀ xβ ) if α contains no free occurrences of x (Ax6) α → β if α is a variant of β (Ax7) ∀ x ( x ≈ x ) (Ax8) ( x ≈ y ) → ( α → α [ x ≀ y ]) if y is a variable free for x in α (N = ) ( x ≈ y ) → (cid:3) ( x ≈ y ) (P = ) ¬ ( x ≈ y ) → (cid:3) ¬ ( x ≈ y ) (K) (cid:3) ( α → β ) → ( (cid:3) α → (cid:3) β ) (K1) (cid:3) ( α → β ) → ( ♦ α → ♦ β ) (K2) ♦ ( α → β ) → ( (cid:3) α → ♦ β ) (M1) (cid:3) ¬ α → (cid:3) ( α → β ) (M2) (cid:3) β → (cid:3) ( α → β ) (M3) ♦ β → ♦ ( α → β ) (M4) ♦ ¬ α → ♦ ( α → β ) (T) (cid:3) α → α (DN1) (cid:3) α → (cid:3) ¬¬ α (DN2) (cid:3) ¬¬ α → (cid:3) α (BF) ∀ x (cid:3) α → (cid:3) ∀ xα (CBF) (cid:3) ∀ xα → ∀ x (cid:3) α (NBF) ∀ x ♦ α → ♦ ∀ xα (PBF) ♦ ∀ xα → ∀ x ♦ α ( MP ) : β follows from α and α → β ( Gen ) : ∀ xα follows from α The notion of derivation in Tm ∗ is defined as usual. We will use the conventional notationΓ ⊢ Tm ∗ α in order to express that there is a derivation in Tm ∗ of α from Γ.As it could be expected, given that the rule of necessitation is not present in Tm ∗ , this logicsatisfies the restricted version of the Deduction metatheorem (DMT), as usually presented infirst-order logics (see [30]): Theorem 2.3 (Deduction Metatheorem (DMT) for Tm ∗ ) . Suppose that there exists in Tm ∗ a derivation of ψ from Γ ∪ { ϕ } , such that no application of the rule (Gen) has, as its quantifiedvariable, a free variable of ϕ (in particular, this holds when ϕ is a sentence). Then Γ ⊢ Tm ∗ ϕ → ψ . Remark 2.4.
It is well-known that normal modal logics admit two different notions of con-sequence relation with respect to Kripke semantics: the local one and the global one (see, for A more general version for quantified classical logic, which also holds for Tm ∗ , can be found in [30, p. 67]. The first-order non-normal modal logic Tm ∗ instance, [6, Defs. 1.35 and 1.37]). According to this, given a class M of Kripke models, aformula ϕ follows locally from a set Γ of formulas if, for any M ∈ M and every world w in M , ϕ is true in h M, w i whenever every formula in Γ is true in h M, w i . In turn, ϕ follows globally from Γ in M if, for any M ∈ M , ϕ is true in h M, w i for every w whenever everyformula in Γ is true in h M, w i for every w . From this, DMT holds in a propositional nor-mal modal logic only for the local (semantical) consequence relation, and this is the perspectiveadopted with most normal modal logics, in which a modal logic can be studied simply in termsof validity of formulas. The same approach is assumed for first-order normal modal logics,ensuring the preservation of the deduction metatheorem. From the proof-theoretical perspective,the preservation of DMT even by considering the generalization inference rule (Gen) and thenecessitation rule forces to redefine the notion of derivation from premises in the correspondingHilbert calculi for such modal logics. In this way, derivations are represented exclusively interms of theoremhood (hence DMT holds by definition). Namely: Γ ⊢ ϕ iff either ⊢ ϕ or thereexist γ , . . . , γ n ∈ Γ such that ⊢ γ → ( γ → ( . . . → ( γ n → ϕ ) . . . )) .Concerning Ivlev-like non-normal modal first-order logics such as Tm ∗ , DMT holds exactlyunder the restrictions imposed in classical first-order logic w.r.t. the application of (Gen) tothe premises. This is a consequence of assuming for Tm ∗ (as well as for the other first-ordersystems to be considered in this paper) the usual notion of derivation from premises in Hilbertcalculi (which, in the case of first-order logics such as classical first-order logic, only works forglobal semantics). Another consequence of discarding the necessitation rule in the Ivlev-like systems is that theReplacement Metatheorem, which holds in every normal modal logic based on (propositional orfirst-order) classical logic, is no longer valid. That is, these logic are not self-extensional. Tm ∗ In this section a suitable semantics of first-order structures for Tm ∗ will be provided, formalizingthe intuitions given in Section 1. As we shall see in Remark 2.6, this semantics follows the linesof the non-deterministic first-order structures for paraconsistent logics based on Nmatrices givenin [13] which, by its turn, is adapted from the standard algebraic approach to first-order logic. Definition 2.5.
Let Θ be a first-order signature. A first-order structure for Tm ∗ is a pair A = h U, · A i , such that U is a non-empty set and · A is an interpretation function for the symbolsof Θ defined as follows: • For each n -ary predicate P , P A = ( a A ( P ) , c A ( P )) such that a A ( P ) ⊆ U n and c A ( P ) ⊆ U n ;it will be required that a A ( ≈ ) = { ( a, a ) : a ∈ U } and c A ( ≈ ) = ∅ ; • For each individual constant c , c A is an element of U ; • For each n -ary function f , f A is a function from U n to U . Remark 2.6.
Within the semantics for first-order languages defined over algebraic structures,a predicate symbol of arity n is interpreted by means of a function I ( P ) : U n → A , where U isthe domain of the semantical structure and A is the domain of a given algebra of truth-values.This generalizes the standard Tarskian structures in which I ( P ) is a subset of U n , which can berepresented by its characteristic function I ( P ) : U n → { , } . This approach was adapted in [13] Of course it would be possible to consider the notion of derivation from premises in the Hilbert calculi inwhich DMT holds by definition, as described above for standard modal logic. The Replacement Metatheorem says that if β is a subformula of α , α ′ is the result of replacing zero ormore occurences of β in α by a wff γ , then: ⊢ β ↔ γ implies that ⊢ α ↔ α ′ (see, for instance [30, p. 72]).In [32] it was shown that Replacement does not hold, in general, in Ivlev-like systems. The first-order non-normal modal logic Tm ∗ for non-deterministic first-order structures for paraconsistent logics based on Nmatrices overmultialgebras called swap structures . In such framework, a predicate symbol P is interpretedas a function I ( P ) : U n → B where B is the domain of a given swap structure. It is easy tosee that the notion of semantical structures for Tm ∗ given in Definition 2.5 is equivalent to theabove mentioned approach. Indeed, let A = h U, · A i be a first-order structure for Tm ∗ . For every n -ary predicate symbol P let P A : U n → V be the function such that P − A ( T + ) = a A ( P ) \ c A ( P ) ; P − A ( C + ) = a A ( P ) ∩ c A ( P ) ; P − A ( C − ) = c A ( P ) \ a A ( P ) ; and P − A ( F − ) = U n \ (cid:0) a A ( P ) ∪ c A ( P ) (cid:1) .This defines a first-order structure as in [13]. Conversely, let A be a first-order structure for Tm ∗ as in [13], that is: it is an structure as in Definition 2.5, with the only difference that the n -ary predicates are interpreted as functions P A : U n → V . Now, define P A = ( a A ( P ) , c A ( P )) such that a A ( P ) = P − A ( T + ) ∪ P − A ( C + ) and c A ( P ) = P − A ( C + ) ∪ P − A ( C − ) . This gives originto a first-order structure for Tm ∗ as in Definition 2.5. Therefore, both semantical approachesare equivalent. Definition 2.7. An assignment over a first-order structure A for Tm ∗ is a function s : V ar → U . Given two assignments s and s ′ over A and a variable x , we say that s and s ′ are x -equivalent , denoted by s ∼ x s ′ , if s ( y ) = s ′ ( y ) for every y ∈ V ar such that y = x . The setof assignments over A will be denoted by A ( A ) . The set of assignments which are x -equivalentto s will be denoted by E x ( s ) . If a ∈ U , the assignment s ′ ∈ E x ( s ) such that s ′ ( x ) = a will bedenoted by s xa . Given A and s , the denotation [[ τ ]] A s of a term τ in ( A , s ) is defined recursively as fol-lows: (i) [[ x ]] A s = s ( x ) if x is a variable; (ii) [[ c ]] A s = c A if c is a constant; and [[ f τ . . . τ n ]] A s = f A ([[ τ ]] A s , . . . , [[ τ n ]] A s ). Note that [[ τ ]] A s ∈ U for every term τ . Definition 2.8.
Given A , a valuation over A is a function v : F or (Θ) × A ( A ) → V definedrecursively as follows:1. For atomic wffs of the form P τ . . . τ n ,- v ( P τ . . . τ n , s ) = T + iff ([[ τ ]] A s , . . . , [[ τ n ]] A s ) ∈ a A ( P ) \ c A ( P ) ;- v ( P τ . . . τ n , s ) = C + iff ([[ τ ]] A s , . . . , [[ τ n ]] A s ) ∈ a A ( P ) ∩ c A ( P ) ;- v ( P τ . . . τ n , s ) = C − iff ([[ τ ]] A s , . . . , [[ τ n ]] A s ) ∈ c A ( P ) \ a A ( P ) ;- v ( P τ . . . τ n , s ) = F − iff ([[ τ ]] A s , . . . , [[ τ n ]] A s ) ∈ U n \ (cid:0) a A ( P ) ∪ c A ( P ) (cid:1) .2. For atomic wffs of the form ( τ ≈ τ ) ,- v (( τ ≈ τ ) , s ) = T + iff [[ τ ]] A s = [[ τ ]] A s ;- v (( τ ≈ τ ) , s ) = F − iff [[ τ ]] A s = [[ τ ]] A s ;3. for wffs of the form α such that ∈ {¬ , (cid:3) } , then v ( α, s ) ∈ ˜ v ( α, s ) , where ˜ denotesthe corresponding multioperation of A Tm associated to as described in Subsection 2.1.4. For wffs of the form ( α → β ) then v (( α → β ) , s ) ∈ v ( α, s ) ˜ → v ( α, s ) , where ˜ → denotesthe multioperation of A Tm as described in Subsection 2.1.5. For wffs of the form ∀ xα , let X ( α, x, v, s ) = { v ( α, s ′ ) : s ′ ∈ E x ( s ) } (recall Definition 2.7).Then v ( ∀ xα, s ) ∈ ˜ ∀ d (cid:0) X ( α, x, v, s ) (cid:1) , where ˜ ∀ d is defined as in Subsection 2.1.6. Let τ be a term free for a variable z in formula ϕ , and let b = [[ τ ]] A s . Then v ( ϕ [ z/τ ] , s ) = v ( ϕ, s zb ) (recalling the notation from Definition 2.7).7. If ϕ and ϕ ′ are variant, then v ( ϕ, s ) = v ( ϕ ′ , s ) for every s . The first-order non-normal modal logic Tm ∗
8. If y is a variable free for x in ϕ then v (( x ≈ y ) → ( ϕ → ϕ [ x ≀ y ]) , s ) ∈ { T + , C + } . Remark 2.9.
Recall the sentences (1)-(3) used in Section 1 as motivating examples. Clause in Definition 2.8 intends to capture formally the informal considerations about the sentence(1) given above. In turn, clause intends to formalize the considerations above about sentence(3). Clause is the formal counterpart of the considerations about the sentence (2).The intuition about clauses and was already discussed in [12]. The reader should ask ifthose non-deterministic operators, specially with respect to the implication operator, are ratherarbitrary. In some sense, that is true and we agree with that criticism. We think, also, that evenIvlev would agree with that. Maybe this justifies that he proposed several different modal systemswith respect to the implication operator. Anyway, the multioperator above for the implication isnot completely arbitrary. In [12] we presented some arguments in order to convince the readerthat Ivlev’s multioperator for Tm implication is reasonably natural from an intuitive point ofview. With respect to the negation, we don’t have many options and we showed there thatthis operator is the most natural in this context. With respect to the semantical counterpartof the modal connective (cid:3) , Ivlev proposed in [23] not only the multioperator described inSubsection 2.1, but also many others. We choose this one because it does not collapse anyiterations of the modal connective (cid:3) . In Subsection 3.2 two cases of collapse of modal iterationswill be discussed.Finally, clauses to are necessary in order to deal within a non-deterministic frameworkin a coherent way, as it was already analyzed in [13] for first-order paraconsistent logics. Forinstance, the substitution lemma (which is crucial in order to validate (Ax4) ) must be guaranteedby requiring to the valuations to choose in a suitable way, and this is stated by clause . Definition 2.10.
Let A be a first-order structure for Tm ∗ , and let v be a valuation over it.Then, ( A , v ) satisfies a wff α if v ( α, s ) ⊆ { T + , C + } for some assignment s over A . The formula α is true in ( A , v ) if it is satisfied by every assignment s . We say that α is valid in Tm ∗ if itis true in every ( A , v ) . Finally, a formula α is a semantical consequence of a set Γ of formulasw.r.t. first-order structures for Tm ∗ , denoted by Γ | = Tm ∗ α , if, for every structure A and everyvaluation v , if every γ ∈ Γ is true in ( A , v ) then α is true in ( A , v ) . Remark 2.11.
The notion of semantical entailment in Tm ∗ deserves some comments. Observethat the standard semantics for first-order classical logic w.r.t. Tarskian structures produces aunique denotation for formulas, given a structure and an assignment. The same holds for first-order structures defined over complete Boolean algebras, or over complete Heying algebras forfirst-order intuitionistic logic, among other examples of first-order algebraizable logics (see, forinstance, the classical references [35, 36]). In the present non-deterministic semantics, it ispossible to make several choices for the denotation of complex formulas in a given structure,which are performed by the valuations. This is why the basic semantical environment for Tm ∗ is given by pairs ( A , v ) , and so the semantical consequence is given by taking assignments oversuch pairs, in order to interpret the free variables. The same situation happens with the non-deterministic semantics for paraconsistent logics presented in [13]. In this section the soundness of Tm ∗ w.r.t. its non-deterministic semantics will be stated. Lemma 2.12.
Let s and s ′ be two assignments over A which are x -equivalent. Then, v ( α, s ) = v ( α, s ′ ) for every formula α in which x does not occur free.Proof. It can be proved easily by induction on the complexity of α . The first-order non-normal modal logic Tm ∗ Lemma 2.13.
Let s be an assignment and v a valuation over a structure A . Then:(1) v ( ∀ xα → α [ x/τ ] , s ) ∈ { T + , C + } if τ is a term free for x in α .(2) v ( ∀ x ( α → β ) → ( α → ∀ xβ ) , s ) ∈ { T + , C + } if α contains no free occurrences of x .(3) v ( ∀ x ( x ≈ x ) , s ) = T + .(4) v (( x ≈ y ) → (cid:3) ( x ≈ y ) , s ) ∈ { T + , C + } (5) v ( ¬ ( x ≈ y ) → (cid:3) ¬ ( x ≈ y ) , s ) ∈ { T + , C + } .(6) v ( ∀ x (cid:3) α → (cid:3) ∀ xα, s ) ∈ { T + , C + } .(7) v ( (cid:3) ∀ xα → ∀ x (cid:3) α, s ) ∈ { T + , C + } .(8) v ( ∀ x ♦ α → ♦ ∀ xα, s ) ∈ { T + , C + } .(9) v ( ♦ ∀ xα → ∀ x ♦ α.s ) ∈ { T + , C + } .Proof. (1) Suppose that v ( ∀ xα, s ) ∈ { T + , C + } . Then { v ( α, s ′ ) : s ′ ∈ E x ( s ) } ⊆ { T + , C + } , byDefinition 2.8(5) and by the definition of ˜ ∀ d given in Subsection 2.1. Let b = [[ τ ]] A s . Then v ( α [ x/τ ] , s ) = v ( α, s xb ), by Definition 2.8(6). Since s xb ∈ E x ( s ) it follows that v ( α, s xb ) ∈{ T + , C + } , by hypothesis. This means that v ( α [ x/τ ] , s ) ∈ { T + , C + } and so v ( ∀ xα → α [ x/τ ] , s ) ∈{ T + , C + } .(2) Assume that v ( ∀ x ( α → β ) , s ) ∈ { T + , C + } and v ( α, s ) ∈ { T + , C + } . As proved in (1), { v ( α → β, s ′ ) : s ′ ∈ E x ( s ) } ⊆ { T + , C + } . Let s ′ ∈ E x ( s ). Then v ( α → β, s ′ ) ∈ { T + , C + } .But v ( α → β, s ′ ) ∈ v ( α, s ′ ) ˜ → v ( β, s ′ ), and v ( α, s ′ ) = v ( α, s ), by Lemma 2.12 (since x doesnot occur free in α ). Then v ( α, s ′ ) ∈ { T + , C + } and so v ( β, s ′ ) ∈ { T + , C + } . From this, { v ( β, s ′ ) : s ′ ∈ E x ( s ) } ⊆ { T + , C + } . This shows that v ( ∀ xβ, s ) ∈ { T + , C + } , proving (2).(3) It is an immediate consequence of the definitions.(4) Suppose that v (( x ≈ y ) , s ) ∈ { T + , C + } . By Definition 2.8(2), v (( x ≈ y ) , s ) = T + and so v ( (cid:3) ( x ≈ y ) , s ) ∈ { T + , C + } , by Definition 2.8(3) and by the definition of ˜ (cid:3) .(5) Suppose that v ( ¬ ( x ≈ y ) , s ) ∈ { T + , C + } . By Definition 2.8(2), v ( ¬ ( x ≈ y ) , s ) = T + and so(as in item (4)) v ( (cid:3) ¬ ( x ≈ y ) , s ) ∈ { T + , C + } .(6) Suppose that v ( ∀ x (cid:3) α, s ) ∈ { T + , C + } . Then { v ( (cid:3) α, s ′ ) : s ′ ∈ E x ( s ) } ⊆ { T + , C + } . Bydefinition of ˜ (cid:3) (and by Definition 2.8(3)) it follows that { v ( α, s ′ ) : s ′ ∈ E x ( s ) } = { T + } .By Definition 2.8(5) and by the definition of ˜ ∀ d it follows that v ( ∀ xα, s ) = T + . From this v ( (cid:3) ∀ xα, s ) ∈ { T + , C + } , which proves (6).(7) Suppose that v ( (cid:3) ∀ xα, s ) ∈ { T + , C + } . Then v ( ∀ xα, s ) = T + and so { v ( α, s ′ ) : s ′ ∈ E x ( s ) } = { T + } . From this, { v ( (cid:3) α, s ′ ) : s ′ ∈ E x ( s ) } ⊆ { T + , C + } , therefore v ( ∀ x (cid:3) α, s ) ∈{ T + , C + } . This proves (7).(8) Suppose that v ( ∀ x ♦ α, s ) ∈ { T + , C + } . Then { v ( ♦ α, s ′ ) : s ′ ∈ E x ( s ) } ⊆ { T + , C + } .By definition of ˜ ♦ (and by Definition 2.8(3)) it follows that F −
6∈ { v ( α, s ′ ) : s ′ ∈ E x ( s ) } .By Definition 2.8(5) and by the definition of ˜ ∀ d it follows that v ( ∀ xα, s ) = F − . From this v ( ♦ ∀ xα, s ) ∈ { T + , C + } , which proves (8).(9) Suppose that v ( ♦ ∀ xα, s ) ∈ { T + , C + } . Then v ( ∀ xα, s ) = F − and so F − / ∈ { v ( α, s ′ ) : s ′ ∈ E x ( s ) } . From this, { v ( ♦ α, s ′ ) : s ′ ∈ E x ( s ) } ⊆ { T + , C + } , therefore v ( ∀ x ♦ α, s ) ∈ { T + , C + } .This concludes the proof. Lemma 2.14.
Let A be a structure, and v a valuation over it. Then:(1) If α and α → β are true in ( A , v ) , so is β .(2) If α is true in ( A , v ) , so is ∀ xα .Proof. It is an easy consequence of the definitions.
Theorem 2.15 (Soundness) . Let Γ ∪ { α } be a set of formulas. Then: Γ ⊢ Tm ∗ α implies that The first-order non-normal modal logic Tm ∗ Γ | = Tm ∗ α .Proof. By definition of valuation, axioms (Ax6) and (Ax8) are true in every ( A , v ). As itwas proved in [12], all the schema axioms from Tm are valid w.r.t. its four-valued Nmatrixsemantics, so they are also valid in Tm ∗ . By Lemma 2.13, the rest of the schema axioms from Tm ∗ (not considered above) are also true in every ( A , v ). By Lemma 2.14, the inference rulesof Tm ∗ preserve trueness in a given structure and valuation. From this, and assuming thatΓ ⊢ Tm ∗ α , it is easy to prove, by induction on the length of a derivation in Tm ∗ of α from Γ,that Γ | = Tm ∗ α . In this section, the completeness of Tm ∗ w.r.t. its structures will be obtained. In order todo this, the completeness proof for classical first-order logic with equality found in [9] will beadapted. This adaptation is similar to the one found in [13] for first-order paraconsistent logics.Other details of the proof will be also based on the proof of completeness of the modal logic Tm given in [10] and [11]. Definition 2.16.
Consider a theory ∆ ⊆ F or (Θ) and a nonempty set C of constants of thesignature Θ . Then, ∆ is called a C - Henkin theory in Tm ∗ if it satisfies the following: forevery formula ψ with (at most) a free variable x , there exists a constant c in C such that ∆ ⊢ Tm ∗ ψ [ x/c ] → ∀ xψ . Definition 2.17.
Let Θ C be the signature obtained from Θ by adding a set C of new individualconstants. The consequence relation ⊢ C Tm ∗ is the consequence relation of Tm ∗ over the signature Θ C . As it happens with first-order classical logic, the following result can be obtained for Tm ∗ : Theorem 2.18.
Every theory ∆ ⊆ F or (Θ) in Tm ∗ over a signature Θ can be conservativelyextended to a C -Henkin theory ∆ H ⊆ F or (Θ C ) in Tm ∗ over a signature Θ C as in Defini-tion 2.17. That is, ∆ ⊆ ∆ H and: ∆ ⊢ Tm ∗ ϕ iff ∆ H ⊢ C Tm ∗ ϕ for every ϕ ∈ F or (Θ) . In addition,if ∆ H ⊆ ∆ H ⊆ F or (Θ C ) then ∆ H is also a C -Henkin theory. Recall (see for instance Part I of this paper [12]) that, given a Tarskian and finitary logic L = h F or, ⊢i (where F or is the set of formulas of L ), and given a set of formulas Γ ∪ { ϕ } ⊆ F or , the set Γ is maximal non-trivial with respect to ϕ (or ϕ -saturated ) in L if the followingholds: (i) Γ ϕ , and (ii) Γ , ψ ⊢ ϕ for every ψ / ∈ Γ. Observe that if Γ is ϕ -saturated then, forevery wff β : Γ ⊢ β iff β ∈ Γ. In addition, for any wff β : either β ∈ Γ or ¬ β ∈ Γ (but not bothsimultaneously).By adapting to Tm ∗ a classical and general result by Lindenbaum and Lo´s (see [42, Theo-rem 22.2] and [12, Theorem 4.8]) we obtain the following: Theorem 2.19.
Let Γ ∪ { ϕ } ⊆ F or (Θ) such that Γ Tm ∗ ϕ . Then, there exists a set offormulas ∆ ⊆ F or (Θ) which is ϕ -saturated in Tm ∗ and such that Γ ⊆ ∆ . Now, a canonical structure will be defined.
Proposition 2.20.
Let ∆ be a set of formulas over a signature Θ which is ϕ -saturated for agiven formula ϕ in Tm ∗ . Suppose that, in addition, ∆ is a C -Henkin theory for a set C ofconstants in Θ . Define over C the following relation: c ∼ d iff ∆ ⊢ Tm ∗ ( c ≈ d ) . Then ∼ is anequivalence relation. The first-order non-normal modal logic Tm ∗ Proof.
It is an immediate consequence of the properties of the identity predicate in Tm ∗ .For every c ∈ C let ˜ c = { d ∈ C : c ∼ d } and U ∆ = { ˜ c : c ∈ C } . Definition 2.21.
Let ∆ , ϕ and C as in Proposition 2.20. The canonical structure A ∆ = h U ∆ , · A ∆ i is such that:(a) For each n -ary predicate P , P A ∆ = ( a A ∆ ( P ) , c A ∆ ( P )) is such that:- (˜ c , . . . , ˜ c n ) ∈ a A ∆ ( P ) iff P c . . . c n ∈ ∆ ;- (˜ c , . . . , ˜ c n ) ∈ c A ∆ ( P ) iff ¬ ( ¬ (cid:3) P c . . . c n → (cid:3) ¬ P c . . . c n ) ∈ ∆ .(b) For each n -ary function symbol, f A ∆ is given by f A ∆ (˜ c , . . . , ˜ c n ) := ˜ c where c ∈ C is suchthat ( f c . . . c n ≈ c ) ∈ ∆ .(c) For each individual constant c , c A ∆ := ˜ d , where d ∈ C is such that ( c ≈ d ) ∈ ∆ . Proposition 2.22.
The canonical structure A ∆ = h U ∆ , · A ∆ i is well-defined.Proof. The proof is an easy adaptation of the one for classical first-order logic, see for in-stance [9, Lemma 2.12]. With respect to the definition of the functions a A ∆ ( P ) and c A ∆ ( P ),it is enough to observe that if ( c i ≈ d i ) ∈ ∆ for 1 ≤ i ≤ n then: ϕ [ x /c , · · · , x n /c n ] ∈ ∆ iff ϕ [ x /d , · · · , x n /d n ] ∈ ∆, for every formula ϕ whose free variables occur in the list x , . . . , x n . Definition 2.23.
Let A ∆ = h U ∆ , · A ∆ i be the canonical structure for a set of formulas ∆ as inDefinition 2.21. The canonical valuation v ∆ over A ∆ is defined as follows, for any assignment s such that s ( x i ) = ˜ c i for every i ≥ :(a) v ∆ ( α, s ) = T + iff (cid:3) α [ x /c , · · · , x n /c n ] ∈ ∆ ;(b) v ∆ ( α, s ) = C + iff α [ x /c , · · · , x n /c n ] ∈ ∆ and ¬ (cid:3) α [ x /c , · · · , x n /c n ] ∈ ∆ ;(c) v ∆ ( α, s ) = C − iff ¬ α [ x /c , · · · , x n /c n ] ∈ ∆ and ¬ (cid:3) ¬ α [ x /c , · · · , x n /c n ] ∈ ∆ ;(d) v ∆ ( α, s ) = F − iff (cid:3) ¬ α [ x /c , · · · , x n /c n ] ∈ ∆ ,where the free variables occurring in α belong to the list x , . . . , x n . Lemma 2.24.
Let A ∆ = h U ∆ , · A ∆ i be the canonical structure for a set of formulas ∆ asin Definition 2.21. Then, v ∆ is a Tm ∗ -valuation over A ∆ . In addition, for every formula β ∈ F or (Θ) such that the variables occurring in it belong to the list x , . . . , x n , and for everyassignment s such that s ( x i ) = ˜ c i for every i ≥ : v ∆ ( β, s ) ∈ { T + , C + } iff β [ x /c , · · · , x n /c n ] ∈ ∆ .Proof. First, observe that v ∆ is a well-defined function. It follows by the same argumentsgiven in the proof of Proposition 2.22, by axiom (T) and by the properties of ϕ -saturated sets,namely: for every wff β , either β ∈ ∆ or ¬ β ∈ ∆ (but not both simultaneously). The factthat, for every β ∈ F or (Θ), v ∆ ( β, s ) ∈ { T + , C + } iff β [ x /c , · · · , x n /c n ] ∈ ∆ (for s ( x i ) = ˜ c i ) isa consequence of the definition of v ∆ and the considerations above. Thus, it suffices to provethat v ∆ satisfies the clauses 1-8 of Definition 2.8. The proof for clauses 1-5 of such definitionwill be done by induction on the complexity of the formula α . Observe that, if τ is a term suchthat the variables occurring in it belong to the list x , . . . , x n and s ( x i ) = ˜ c i for every i ≥ ∗ ) [[ τ ]] A ∆ s = ˜ c iff ( τ [ x /c , · · · , x n /c n ] ≈ c ) ∈ ∆ . The first-order non-normal modal logic Tm ∗ Clause 1: α is an atomic formula of the form P τ . . . τ k . Suppose that the variables occurringfree in α belong to the list x , . . . , x n and s ( x i ) = ˜ c i for every i ≥
1. Let [[ τ i ]] A ∆ s = ˜ d i for1 ≤ i ≤ k .(a) Suppose that v ∆ ( P τ . . . τ k , s ) = T + . Thus, ( (cid:3) P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆.By (T) , ( P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆. By ( ∗ ), P d . . . d k ∈ ∆. By Defini-tion 2.21(a), ( ˜ d , . . . , ˜ d k ) ∈ a A ∆ ( P ) and so ([[ τ ]] A ∆ s , . . . , [[ τ k ]] A ∆ s ) ∈ a A ∆ ( P ). Besides,( (cid:3) α → ( ¬ (cid:3) α → β ))[ x /c , · · · , x n /c n ] ∈ ∆, since it is an instance of a theoremof classical logic. Then ( ¬ (cid:3) P τ . . . τ k → (cid:3) ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆, by(MP). So, ¬ ( ¬ (cid:3) P τ . . . τ k → (cid:3) ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] / ∈ ∆, by ∆-maximali-ty. By reasoning as above, we conclude that ([[ τ ]] A ∆ s , . . . , [[ τ k ]] A ∆ s ) / ∈ c A ∆ ( P ).Conversely, suppose that ([[ τ ]] A ∆ s , . . . , [[ τ k ]] A ∆ s ) ∈ a A ∆ ( P ) \ c A ∆ ( P ). By reasoningas above, ( P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆. Since ([[ τ ]] A ∆ s , . . . , [[ τ k ]] A ∆ s ) / ∈ c A ∆ ( P )then, by ∆-maximality, ( ¬ (cid:3) P τ . . . τ k → (cid:3) ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆. Sup-pose now that ( ¬ (cid:3) P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆. From this we conclude, by(MP), that ( (cid:3) ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆. But then, as a consequence ofaxiom (T) we get that ( ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆ and so ∆ would beinconsistent. Thus, ( ¬ (cid:3) P τ . . . τ k )[ x /c , · · · , x n /c n ] / ∈ ∆ and so, by maximality,( (cid:3) P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆. Therefore, v ∆ ( P τ . . . τ k , s ) = T + .(b) Observe that v ∆ ( P τ . . . τ k , s ) = C + iff( P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆ and ( ¬ (cid:3) P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆ . In this case ([[ τ ]] A ∆ s , . . . , [[ τ k ]] A ∆ s ) ∈ a A ∆ ( P ). Suppose now, by absurd, that( ¬ (cid:3) P τ . . . τ k → (cid:3) ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆ . By (MP), it would follows that ( (cid:3) ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆. By (T) , wewould have ( ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆ and so ∆ would be inconsistent.From that, we infer that ( ¬ (cid:3) P τ . . . τ k → (cid:3) ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] / ∈ ∆ and,so, by ∆-maximality, ¬ ( ¬ (cid:3) P τ . . . τ k → (cid:3) ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆. Thus,([[ τ ]] A ∆ s , . . . , [[ τ k ]] A ∆ s ) ∈ c A ∆ ( P ).Conversely, suppose now that ([[ τ ]] A ∆ s , . . . , [[ τ k ]] A ∆ s ) ∈ a A ∆ ( P ) ∩ c A ∆ ( P ). By rea-soning as in the previous cases, it follows that ( P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆and ¬ ( ¬ (cid:3) P τ . . . τ k → (cid:3) ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆. Now, suppose that( (cid:3) P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆. But( (cid:3) P τ . . . τ k → ( ¬ (cid:3) P τ . . . τ k → (cid:3) ¬ P τ . . . τ k ))[ x /c , · · · , x n /c n ] ∈ ∆ , since it is an instance of a theorem of classical propositional logic. Thus, by (MP),we would have that ( ¬ (cid:3) P τ . . . τ k → (cid:3) ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆ and so ∆would be inconsistent. Hence, ( (cid:3) P τ . . . τ k )[ x /c , · · · , x n /c n ] / ∈ ∆, and so( ¬ (cid:3) P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆ , by ∆-maximality. Therefore, v ∆ ( P τ . . . τ k , s ) = C + .(c) Note that v ∆ ( P τ . . . τ k , s ) = C − iff( ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆ and ( ¬ (cid:3) ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆ . So (
P τ . . . τ k )[ x /c , · · · , x n /c n ] / ∈ ∆, by ∆-maximality. Hence, ([[ τ ]] A ∆ s , . . . , [[ τ k ]] A ∆ s ) / ∈ a A ∆ ( P ). Suppose now that ( ¬ (cid:3) P τ . . . τ k → (cid:3) ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆.As ( ¬ δ → β ) → ( ¬ β → δ ) is a theorem of classical logic, we have that( ¬ (cid:3) ¬ P τ . . . τ k → (cid:3) P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆ . The first-order non-normal modal logic Tm ∗ Hence, ( (cid:3)
P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆. Thus, ( P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆ by (T) , and so ∆ would be inconsistent. Therefore,( ¬ (cid:3) ¬ P τ . . . τ k → (cid:3) ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] / ∈ ∆ . We conclude that ¬ ( ¬ (cid:3) P τ . . . τ k → (cid:3) ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆, by ∆-maximality, and so ([[ τ ]] A ∆ s , . . . , [[ τ k ]] A ∆ s ) ∈ c A ∆ ( P ).Conversely, suppose that ([[ τ ]] A ∆ s , . . . , [[ τ k ]] A ∆ s ) ∈ c A ∆ ( P ) \ a A ∆ ( P ). By reasoningas in the previous cases, it follows that ( P τ . . . τ k )[ x /c , · · · , x n /c n ] / ∈ ∆ and ¬ ( ¬ (cid:3) ¬ P τ . . . τ k → (cid:3) ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆. By ∆-maximality,( ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆ . If ( (cid:3) ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆, then we would have that ( ¬ (cid:3) ¬ P τ . . . τ k → (cid:3) ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆ and so ∆ woud be inconsistent. Hence,( (cid:3) ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] / ∈ ∆and so, by ∆-maximality, ( ¬ (cid:3) ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆. Therefore, weconclude v ∆ ( P τ . . . τ k , s ) = C − .(d) v ∆ ( P τ . . . τ k , s ) = F − iff ( (cid:3) ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆. Thus, by (T) ,( ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆ and ( P τ . . . τ k )[ x /c , · · · , x n /c n ] / ∈ ∆, by ∆-maximality. Therefore ([[ τ ]] A ∆ s , . . . , [[ τ k ]] A ∆ s ) / ∈ a A ∆ ( P ). By (Ax1) and (MP),( ¬ (cid:3) P τ . . . τ k → (cid:3) ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆ . Since ∆ is consistent, ¬ ( ¬ (cid:3) P τ . . . τ k → (cid:3) ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] / ∈ ∆. Fromthis, it follows that ([[ τ ]] A ∆ s , . . . , [[ τ k ]] A ∆ s ) / ∈ c A ∆ ( P ).Suppose now that ([[ τ ]] A ∆ s , . . . , [[ τ k ]] A ∆ s ) / ∈ a A ∆ ( P ) ∪ c A ∆ ( P ). As above, it is concludedthat ( P τ . . . τ k )[ x /c , · · · , x n /c n ] / ∈ ∆ and also ¬ ( ¬ (cid:3) P τ . . . τ k → (cid:3) ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] / ∈ ∆ . Thus ( ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆ and( ¬ (cid:3) P τ . . . τ k → (cid:3) ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆ , by ∆-maximality. If ( (cid:3) P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆, then( P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆ , by (T) , and so ∆ would be inconsistent. Thus ( (cid:3) P τ . . . τ k )[ x /c , · · · , x n /c n ] / ∈ ∆and so, by ∆-maximality, ( ¬ (cid:3) P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆. Hence, by (MP),( (cid:3) ¬ P τ . . . τ k )[ x /c , · · · , x n /c n ] ∈ ∆. Therefore, v ∆ ( P τ . . . τ k , s ) = F − . Clause 2: α is an atomic formula of the form ( τ ≈ τ ). As in Clause 1 , suppose that thevariables occurring in τ and τ belong to the list x , . . . , x n and s ( x i ) = ˜ c i for every i ≥
1. Let [[ τ i ]] A ∆ s = ˜ d i for 1 ≤ i ≤ v ∆ (( τ ≈ τ ) , s ) = T + . Thus, ( (cid:3) ( τ ≈ τ ))[ x /c , · · · , x n /c n ] ∈ ∆and so, by (T) , ( τ ≈ τ )[ x /c , · · · , x n /c n ] ∈ ∆. Then, ( d ≈ d ) ∈ ∆ and so[[ τ ]] A ∆ s = [[ τ ]] A ∆ s .Conversely, suppose that [[ τ ]] A ∆ s = [[ τ ]] A ∆ s . Then, ( d ≈ d ) ∈ ∆ and so ( τ ≈ τ )[ x /c , · · · , x n /c n ] ∈ ∆. Then, by (N = ) , ( (cid:3) ( τ ≈ τ ))[ x /c , · · · , x n /c n ] ∈ ∆.Thus, v ∆ (( τ ≈ τ ) , s ) = T + . The first-order non-normal modal logic Tm ∗ (b) Suppose that v ∆ (( τ ≈ τ ) , s ) = F − . Thus, ( (cid:3) ¬ ( τ ≈ τ ))[ x /c , · · · , x n /c n ] ∈ ∆.By (T) , ( ¬ ( τ ≈ τ ))[ x /c , · · · , x n /c n ] ∈ ∆. As ∆ is nontrivial,( τ ≈ τ )[ x /c , · · · , x n /c n ] / ∈ ∆ . Hence, ( d ≈ d ) / ∈ ∆ and so [[ τ ]] A ∆ s = [[ τ ]] A ∆ s .Suppose now that [[ τ ]] A ∆ s = [[ τ ]] A ∆ s . Thus, ( τ ≈ τ )[ x /c , · · · , x n /c n ] / ∈ ∆ and so( ¬ ( τ ≈ τ ))[ x /c , · · · , x n /c n ] ∈ ∆, by ∆-maximality. Then, by (P = ) , ( (cid:3) ¬ ( τ ≈ τ ))[ x /c , · · · , x n /c n ] ∈ ∆. In other words, v ∆ (( τ ≈ τ ) , s ) = F − . Clauses 3-4: α is of the form ¬ β , (cid:3) β or β → γ . The proof is analogous to the one for Tm given in [10, Lemma 4] (see also [11, Lemma 4]). Clause 5: α is of the form ∀ xβ . As above, suppose that the variables occurring free in α belongto the list x , . . . , x n and s ( x i ) = ˜ c i for every i ≥
1. Assume, without loss of generality,that x = x i for 1 ≤ i ≤ n .(a) Suppose that v ∆ ( ∀ xβ, s ) = T + . Then, ( (cid:3) ∀ xβ )[ x /c , · · · , x n /c n ] ∈ ∆. By (CBF) and (MP) we have that ( ∀ x (cid:3) β )[ x /c , · · · , x n /c n ] ∈ ∆. Thus, by (Ax4) , we have that( (cid:3) β )[ x /c , · · · , x n /c n ][ x/c ] ∈ ∆, for every c ∈ C . This means that v ∆ ( β, s ′ ) = T + for every s ′ ∈ E x ( s ). Then, v ∆ ( ∀ xβ, s ) ∈ ˜ ∀ d (cid:0) X ( β, x, v ∆ , s ) (cid:1) , where the notation is asin Definition 2.8(5).(b) Suppose that v ∆ ( ∀ xβ, s ) = C + . Thus, ( ∀ xβ )[ x /c , · · · , x n /c n ] ∈ ∆ and, in addition,( ¬ (cid:3) ∀ xβ )[ x /c , · · · , x n /c n ] ∈ ∆. By (Ax4) , β [ x /c , · · · , x n /c n ][ x/c ] ∈ ∆ for every c ∈ C . This means that v ∆ ( β, s ′ ) ⊆ { T + , C + } for every s ′ ∈ E x ( s ). If v ∆ ( β, s ′ ) = T + for every s ′ ∈ E x ( s ) then ( (cid:3) β )[ x /c , · · · , x n /c n ][ x/c ] ∈ ∆, for every c ∈ C . Let c ∈ C such that ∆ ⊢ Tm ∗ ψ [ x/c ] → ∀ xψ , for ψ = ( (cid:3) β )[ x /c , · · · , x n /c n ]. Fromthis, ( ∀ x (cid:3) β )[ x /c , · · · , x n /c n ] ∈ ∆. By (BF) , ( (cid:3) ∀ xβ )[ x /c , · · · , x n /c n ] ∈ ∆, hence∆ would be trivial. This shows that v ∆ ( β, s ′ ) = C + for some s ′ ∈ E x ( s ). Then, v ∆ ( ∀ xβ, s ) ∈ ˜ ∀ d (cid:0) X ( β, x, v ∆ , s ) (cid:1) , where the notation is as in Definition 2.8(5).(c) Suppose that v ∆ ( ∀ xβ, s ) = C − . Then,( ¬∀ xβ )[ x /c , · · · , x n /c n ] ∈ ∆ and( ¬ (cid:3) ¬∀ xβ )[ x /c , · · · , x n /c n ] ∈ ∆ . By (PBF) , ( ∀ x ¬ (cid:3) ¬ β )[ x /c , · · · , x n /c n ] ∈ ∆. Since ∆ is C -Henkin,( ¬ β )[ x /c , · · · , x n /c n ][ x/c ′ ] ∈ ∆for some c ′ ∈ C . On the other hand, ( ¬ (cid:3) ¬ β )[ x /c , · · · , x n /c n ][ x/c ] ∈ ∆ for ev-ery c ∈ C , by (Ax4) . In particular, ( ¬ (cid:3) ¬ β )[ x /c , · · · , x n /c n ][ x/c ′ ] ∈ ∆. Thismeans that v ∆ ( β, s ′ ) = C − for some s ′ ∈ E x ( s ) (namely, for s ′ = s xa with a = ˜ c ′ ).If v ∆ ( β, s ′′ ) = F − for some s ′′ ∈ E x ( s ) then ( (cid:3) ¬ β )[ x /c , · · · , x n /c n ][ x/c ′′ ] ∈ ∆for some c ′′ ∈ C , hence ∆ would be trivial. Therefore v ∆ ( β, s ′′ ) = F − for every s ′′ ∈ E x ( s ) and so v ∆ ( ∀ xβ, s ) ∈ ˜ ∀ d (cid:0) X ( β, x, v ∆ , s ) (cid:1) , where the notation is as in Defi-nition 2.8(5).(d) Suppose that v ∆ ( ∀ xβ, s ) = F − . Thus,( ¬∀ xβ )[ x /c , · · · , x n /c n ] ∈ ∆ and( (cid:3) ¬∀ xβ )[ x /c , · · · , x n /c n ] ∈ ∆ . The first-order non-normal modal logic Tm ∗ As ( γ → ¬ δ ) → ( δ → ¬ γ ) is a theorem of classical logic we have, by (NBF) , that( ¬∀ x ¬ (cid:3) ¬ β )[ x /c , · · · , x n /c n ] ∈ ∆. Since ∆ is a C -Henkin theory and ( ¬ γ → δ ) → ( ¬ δ → γ ) is a theorem of classical logic, it follows that( (cid:3) ¬ β )[ x /c , · · · , x n /c n ][ x/c ′ ] ∈ ∆for some constant c ′ in C . This means that v ∆ ( β, s ′ ) = F − for some s ′ ∈ E x ( s )(namely, for s ′ = s xa with a = ˜ c ′ ). Therefore v ∆ ( ∀ xβ, s ) ∈ ˜ ∀ d (cid:0) X ( β, x, v ∆ , s ) (cid:1) , wherethe notation is as in Definition 2.8(5). Clause 6: (Substitution) Let τ be a term free for a variable z in a formula ψ , and let b = ˜ c =[[ τ ]] A ∆ s . By (Ax8) (the Leibinz rule), v ∆ ( ψ [ z/τ ] , s ) = v ∆ ( ψ [ z/c ] , s ) = v ∆ ( ψ, s zb ). Clause 7: (Variant) Let ψ and ψ ′ two formulas such that their free variables belong to thelist x , . . . , x n , and s ( x i ) = ˜ c i for every i ≥
1. Observe that, if ψ and ψ ′ are vari-ant, so are ψ [ x /c , · · · , x n /c n ] and ψ ′ [ x /c , · · · , x n /c n ], as well as the pairs of formulas( ψ )[ x /c , · · · , x n /c n ] and ( ψ ′ )[ x /c , · · · , x n /c n ], for ∈ {¬ , (cid:3) , ¬ (cid:3) , (cid:3) ¬ , ¬ (cid:3) ¬} . Onthe other hand, if ψ and ψ ′ are variant then ψ ∈ ∆ iff ψ ′ ∈ ∆, by (Ax6) . From this, v ∆ ( ψ, s ) = v ∆ ( ψ ′ , s ) and so v ∆ satisfies clause 7 of Definition 2.8. Clause 8: (Leibniz rule) Suppose that y is a variable free for x in a formula ψ . By (Ax8) ,( x ≈ y ) → ( ψ → ψ [ x ≀ y ]) ∈ ∆. Therefore v ∆ (( x ≈ y ) → ( ψ → ψ [ x ≀ y ]) , s ) ∈ { T + , C + } .This concludes the proof. Theorem 2.25 (Completeness) . Let Γ ∪ { α } ⊆ F or (Θ) be a set of formulas. Then: Γ | = Tm ∗ ϕ implies that Γ ⊢ Tm ∗ ϕ .Proof. Let Γ ∪ { ϕ } ⊆ F or (Θ) such that Γ Tm ∗ ϕ . Then, by Theorem 2.18, there exists a C -Henkin theory ∆ H over Θ C in Tm ∗ for a nonempty set C of new individual constants such thatΓ ⊆ ∆ H and, for every α ∈ F or (Θ): Γ ⊢ Tm ∗ α iff ∆ H ⊢ C Tm ∗ α . From this, ∆ H C Tm ∗ ϕ . Then,by Theorem 2.19, there exists a set of formulas ∆ H in Θ C extending ∆ H which is ϕ -saturatedin Tm ∗ . By Theorem 2.18, ∆ H is also a C -Henkin theory over Θ C in Tm ∗ .Consider now the canonical structure A ∆ H for ∆ H over signature Θ C , as in Definition 2.21,and let v ∆ H be the canonical valuation over A ∆ H as in Definition 2.23. Let s be an assignmentover A ∆ H , and let s ( x i ) = ˜ c i for every i ≥
1. If γ ∈ Γ such that the variables occurringin it belong to the list x , . . . , x n then, by (Gen) and by (Ax4) , γ [ x /c , · · · , x n /c n ] ∈ ∆.From this, v ∆ H ( γ, s ) ∈ { T + , C + } . This shows that every γ ∈ Γ is true in ( A ∆ H , v ∆ H ). Onthe other hand, ϕ / ∈ ∆ H . If ϕ is a closed formula then ϕ is not true in ( A ∆ H , v ∆ H ), bythe last assertion of Lemma 2.24. Otherwise, let n = M ax { i : x i occurs free in ϕ } . Let ψ = ∀ x · · · ∀ x n − ϕ . By (Ax4) , ∀ x n ψ ∆ H . Since ∆ H is a C -Henkin theory, there exists c n ∈ C such that ψ [ x n /c n ] ∆ H . Let ψ = ∀ x · · · ∀ x n − ϕ [ x n /c n ]. By reasoning as above,there exists c n − ∈ C such that ψ [ x n − /c n − ] ∆ H . Inductively, it can be proven that thereare constants c , . . . , c n ∈ C such that ϕ [ x /c , · · · , x n /c n ] / ∈ ∆. Let s be an assignmentsuch that s ( x i ) = ˜ c i for every 1 ≤ i ≤ n . Then, by the last assertion of Lemma 2.24, v ∆ H ( ϕ, s ) / ∈ { T + , C + } , showing that ϕ is not true in ( A ∆ H , v ∆ H ). Now, let A be the reduct of A ∆ H to signature Θ, and let v : F or (Θ) × A ( A ) → V be the restriction of v ∆ H to F or (Θ), thatis: v ( α, s ) = v ∆ H ( α, s ) for every α ∈ F or (Θ) and every s in A ( A ) = A ( A ∆ H ). Clearly, v is avaluation over A and s is an assignment over it such that v ( ϕ, s ) / ∈ { T + , C + } , hence ϕ is nottrue in ( A , v ) (as noted above, if ϕ is a closed formula then s can be arbitrary), while every γ ∈ Γ is true in ( A , v ). This shows that Γ = Tm ∗ ϕ . Philosophical questions concerning first-order modal logic In this section a brief philosophical discussion will be considered concerning first-order modallanguages, analyzing the rˆole that the new semantics proposed in the previous sections canplay with respect to this debate. In particular, the problem of the contingent identities and thedistinction between de re and de dicto modalities will be discussed. Of course we do not defendthat the modal non-deterministic semantic presented here can solve those problems: we justwant to invite the reader to rethink them from a different perspective. We are aware that thephilosophical discussion about quantified modal logics is deep and rich: in this paper, whichis mostly of technical character, we are just scratching the surface concerning these complexphilosophical issues. A more detailed conceptual discussion is left for a future work.
We saw that, in standard Kripkean models, identities involving proper names are always neces-sary, since they are rigid designators . But Kripke thesis is stronger than that: he also defendsthat, differently from proper names, identities involving definite descriptions are contingent. Consider, for instance, Kripke’s example:(4) Richard Nixon is the winner of the presidential elections of the United States in 1968.The sentence (4) is an identity involving a proper name and a definite description. Kripkeargues that, although (4) is true, it might be false. Indeed, Richard Nixon could have lost thepresidential elections. In contrast, Richard Nixon is, necessarily, Richard Nixon.The relation between proper names and definite description is not as simple as Kripkeargues. As showed by Ruth C. Barcan, the following sentence is a theorem of any Kripke’sfirst-order modal system: ( x ≈ y ) → (cid:3) ( x ≈ y )Let f be the function for “the winner of the presidential elections of x in 1968”; let c bethe constant for “Richard Nixon”; and let c be the constant for “United States”. Thus, (4)can be symbolized by: (4’) ( c ≈ f c )Therefore, by the theorem above and (MP), we infer from (4’) the following: (cid:3) ( c ≈ f c )This force us to accept that, in Kripkean systems, not only equalities involving propernames, but also those involving definite descriptions are necessary too, against Kripke’s ownthesis. Indeed, the fact that x and y may be replaced by any term makes the distinctionbetween rigid designator (that is, constants) and definite description (that is, functions) totallyirrelevant.The point here is that some kind of equalities seems to be contingent from an intuitiveperspective, as pointed out by Hughes and Cresswell. That is the case of the sentence:(5) The person who lives next door is the mayor. In [29, p. 41]. It was first shown in [5] that it holds for Lewis system S2 and S4 , but is easy to check that it holds in anyKripkean normal modal system. See footnote 9.
Philosophical questions concerning first-order modal logic Surely, (5) is contingent, since it is clearly possible that the person who lives next doormight not have been the mayor. However, according to the theorem above, if (5) is true, thenit is necessarily true.There are two ways, however, to overcome this difficulty. The first one is suggested byKripke himself and it is inspired on Russellian distinction between wider and narrow scope. First of all, we define the uniqueness quantification as follows: ∃ ! xα ≡ def ∃ x ( α ∧ ∀ y ( α [ x/y ] → ( x ≈ y )))Let P be the predicate “ x is a person who lives next door” and P be “ x is a mayor”. Thus,(5) is symbolized by: (5’) ∃ ! xP x ∧ ∃ ! xP x ∧ ∀ x ( P x → P x )In this interpretation, (5’) can be clearly contingently true in Kripkean models. Consider,for instance, the following model: in the world w the individual d is the only one who is inthe extension of P and the only one who is in the extesion of P . But in the world w relatedto w , the individual d is the only one who is in the extension of P , the individual d is theonly one who is in the extension of P and d = d . Clearly, the sentence (5’) is true in w butfalse in w . Therefore, (5’) is contingently true in w .Analogously, let P be the predicate “ x is winner of the presidential election of the UnitedStates in 1968”. Thus, (4) can be formalized by:(4”) ∃ ! x ( P x ∧ ( x ≈ c ))Consider, now, the following model: c is the name of the individual d ; in the world w ,the individual d is the only one who is in the extension of P . But in the world w related to w , the individual d is the only one who is in the extension of P and d = d . Clearly, thesentence (4”) is true in w but false in w . Therefore, (4”) is contingently true in w .The second path is through the intuitive Carnapian notion of intensional objects . From amore contemporaneous perspective, we could understand Carnapian proposal as follows. Letus consider constant domains semantics, for instance. Those models will be enriched by a set I of allowable intensional objects, such that an intensional object i is a function such that foreach possible word w , i ( w ) is a member of the domain, saying that the intensional object is in w . Thus, if “the person who lives next door” is the function i and “the mayor” is the function i ′ , we can say that (5) is contingently true because i ( w ) = i ′ ( w ), but it could be at least someword w ′ related to w such that i ( w ′ ) = i ′ ( w ′ ). Analogously, if “the winner of the presidentialelection of the United States in 1968” is the function i and d is the individual refereed by theconstant c , we can say that (4) is contingently true because i ( w ) = d , but it could be at leastsome word w ′ related to w such that i ( w ′ ) = d . In Kripke’s words: “someone might say that the man who taught Alexander might not have taught Alexan-der; though it could not have been true that: the man who taught Alexander didn’t teach Alexander. This isRussell’s distinction of scope.” See [29, p. 62]. With respect to definite description and Russell’s distinction ofscope, see [39]. Russell uses the expressions “primary occurrence” and “secondary occurrence”. See [20, p. 318–322]. Carnap in [7, p. 1–39] did not use the expression “intensional object”. He said that terms and predicatesof the language (that he calls “predicators”) have an intension and an extension. While “rational animal”and “feather biped” have the same extension, that is, hold for the same individuals, they don’t have thesame meaning and, thus, they are different with respect to the intension. In contrast, “rational animal’ and“human” have the same intension and the same extension too. The intension of an individual constant is calledindividual concept. Predicators and individual constants are extensionally equivalent if they are equivalent insome description state, while the are intensionally equivalent if they are equivalent in all description state. Aspointed out by Carnap, the notion of intension tries to capture the Fregean notion of sense in [18]. We are following here [20, p. 344–347].
Philosophical questions concerning first-order modal logic In contrast, in Ivlev semantics we can deal with contingent identities without taking any ofthose paths. For that, let us consider the following axioms: (C = ¬ (cid:3) ( x ≈ y ) (C = ¬ (cid:3) ¬ ( x ≈ y )Let call Tm ∗ c the system obtained replacing (N = ) and (P = ) with (C = and (C = . It isclear that axioms (C = and (C = are considered for the purpose of capturing the notion ofcontingent identities. From a semantical point of view, we change in Definition 2.8 clause by the following:- v (( τ ≈ τ ) , s ) = C + iff [[ τ ]] A s = [[ τ ]] A s ;- v (( τ ≈ τ ) , s ) = C − iff [[ τ ]] A s = [[ τ ]] A s .With few adjustments in Lemma 2.24, it is easy to obtain completeness for Tm ∗ c . Thus, theproof for this case now runs as follows (recalling the notation stated in the proof of Lemma 2.24):(a) Suppose that v ∆ (( τ ≈ τ ) , s ) = C + . Thus, ( τ ≈ τ )[ x /c , · · · , x n /c n ] ∈ ∆ and so( d ≈ d ) ∈ ∆. From this, [[ τ ]] A ∆ s = [[ τ ]] A ∆ s .Conversely, suppose that [[ τ ]] A ∆ s = [[ τ ]] A ∆ s . Then, ( d ≈ d ) ∈ ∆ and so( τ ≈ τ )[ x /c , · · · , x n /c n ] ∈ ∆ . By (C = , ( ¬ (cid:3) ( τ ≈ τ ))[ x /c , · · · , x n /c n ] ∈ ∆. Thus, v ∆ (( τ ≈ τ ) , s ) = C + .(b) Suppose that v ∆ (( τ ≈ τ ) , s ) = C − . Thus, ( ¬ ( τ ≈ τ ))[ x /c , · · · , x n /c n ] ∈ ∆. As ∆ isnontrivial, ( τ ≈ τ )[ x /c , · · · , x n /c n ] / ∈ ∆. Hence, ( d ≈ d ) / ∈ ∆ and so [[ τ ]] A ∆ s = [[ τ ]] A ∆ s .Suppose now that [[ τ ]] A ∆ s = [[ τ ]] A ∆ s . Thus, ( τ ≈ τ )[ x /c , · · · , x n /c n ] / ∈ ∆ and so ( ¬ ( τ ≈ τ ))[ x /c , · · · , x n /c n ] ∈ ∆, by ∆-maximality. By (C = ,( ¬ (cid:3) ¬ ( τ ≈ τ ))[ x /c , · · · , x n /c n ] ∈ ∆ . In other words, v ∆ (( τ ≈ τ ) , s ) = C − .The considerations above are enough to realize that, in Ivlev semantics, the problem of con-tingent identities is independent of both the Russellian distinction between narrow/wider scopeand the Carnapian distinction between intentional/extensional objects. From a philosophicalpoint of view, this means that it is possible to support that there are contingent identities with-out being compromised with the Russellian theory of definite description or with the Carnapianconcept of intentional objects.The reader may be asking why we should choose between a modal logic where the identitiesare always necessary and another one where all them are always contingent. In fact, it seemsthat identities like those of sentence (5) are contingent, while identities in an arithmeticalcontext, for example, are always necessary. Indeed, such a radical choice is unnecessary. Wecan just use two different symbols, one for each kind of identity relation. For instance, if ≈ is the identity symbol for necessary identities, the symbol ≈ c could be used for contingentidentities. In this case, we should add to Tm ∗ the versions of (Ax7) , (Ax8) , (C = and (C = using the new symbol ≈ c . Mathematical statement could be formalized with ≈ , while factualor empirical identities (or, in Kripkean theory, identities involving some non-rigid designator)should be formalized using ≈ c .The reader might argue that the option of defining two types of equality is also available inpossible worlds semantics. That’s true. However, besides considering two different equalities ≈ Philosophical questions concerning first-order modal logic and ≈ c , in Ivlev semantics we can define a third one with the following meaning: two individualsare equal if and only if necessarily they are identical (in the first sense stated above). For this,we define: τ ≅ τ := (cid:3) ( τ ≈ τ ) . Semantically, this produces the following:- v (( τ ≅ τ ) , s ) ∈ { T + , C + } iff [[ τ ]] A s = [[ τ ]] A s ;- v (( τ ≅ τ ) , s ) ∈ { C − , F − } iff [[ τ ]] A s = [[ τ ]] A s .Observe that this non-deterministic possibility is not available in the possible worlds seman-tics, since Krikpe’s models are deterministic. This turns evident the wide expressive power ofthe present non-deterministic semantical framework for first-order modal logics. Let’s recall the following definition presented in subsection 2.1: ♦ α := ¬ (cid:3) ¬ α There are two simple ways to obtain new first-order alethical systems that extend Tm ∗ . First,we can add axioms in order to iterate (cid:3) and ♦ . This alternative, from a semantical point ofview, means constraining the set of values of a non-deterministic matrix that interprets (cid:3) .We will consider, for instance, two ways to iterate the modal operators by means of thefollowing axioms: (4) (cid:3) α → (cid:3)(cid:3) α (5) ♦(cid:3) α → (cid:3) α In order to validate axioms (4) and (5) , the valuation functions of Definition 2.8 shouldpreserve, respectively the following multioperators: (4) (5) x ˜ (cid:3) x ˜ ♦ xT + { T + } { T + } C + { C − , F − } { T + , C + } C − { C − , F − } { T + , C + } F − { C − , F − } { C − , F − } x ˜ (cid:3) x ˜ ♦ xT + { T + } { T + } C + { F − } { T + } C − { F − } { T + } F − { F − } { F − } We saw in the last subsection that there are two different ways of dealing with equalities.From that and from the two axioms presented above, we can define four new systems:-
T4m ∗ = Tm ∗ ∪ { (4) } - T45m ∗ = T4m ∗ ∪ { (5) } - T4m ∗ c = Tm ∗ c ∪ { (4) } - T45m ∗ c = T4m ∗ c ∪ { (5) } Again, with few adjustments in Lemma 2.24, we can obtain the completeness results forany of those four new systems. In fact, for each kind of equality, we can generate 14 differentmodal systems. The details on the completeness proof are purely technical and not so difficultto obtain. Based on [10] and [11]. Keeping the (N)matrices of negation and implication fixed, as argued in [12].
Philosophical questions concerning first-order modal logic Kneale draws our attention to the fact that the formulas ♦ ∀ xα and ∃ x (cid:3) α are stronger than ∀ x ♦ α and (cid:3) ∃ xα , respectively. In fact, considering S5 ∗ as the constant domain expansion ofthe propositional modal system S5 , we have: ♦ ∀ xα (cid:15) S5 ∗ ∀ x ♦ α but ∀ x ♦ α S5 ∗ ♦ ∀ xα ∃ x (cid:3) α (cid:15) S5 ∗ (cid:3) ∃ xα but (cid:3) ∃ xα S5 ∗ ∃ x (cid:3) α . The author defends that the difference between ∃ x (cid:3) α and (cid:3) ∃ xα is the formal counterpartof Abelardus’ distinction between necessity de rebus and necessity de sensu . This distinctionwas found later on by Peter of Spain as the modal statements de re and de dicto .According to Kneale, Abelard says that assertions of necessity de rebus and de sensu seemto entail each other. It is interesting to note that this philosophical position attributed toAbelard cannot be expressed in Kripke’s framework. The reason is that, in S5 ∗ , none of suchnecessities entails each other. But S5 ∗ models are the strongest ones within this framework, inthe sense that if two formulas are logically independent w.r.t. those models, so they are alsoindependent in any other Kripkean model.The most astonishing result showed by Tich´y in [41] is that the distinction between de re and de dicto formulas is not eliminable, even by supplementing S5 with the axiom schemaexpressing the so-called Principle of Predication of von Wright, as cited below: (PP)
Properties divide into two types: those whose belonging to an object is always eithernecessary or impossible and those whose belonging to an object is always contingent.This principle is symbolized by Tich´y as follows (see [41, p. 391]):(
P P ∗ ) ∀ x ( (cid:3) α ( x ) ∨ (cid:3) ¬ α ( x )) ∨ ∀ x ( ♦ α ( x ) ∧ ♦ ¬ α ( x ))for any wff α ( x ) in which x is the unique variable (possibly) occurring free. Tich´y proved that de re formulas are not eliminable in S5 ∗ + ( P P ∗ ). As a consequence of this, since S5 ∗ +( P P ∗ ) is stronger than any Kripkean first-order standard semantics, Tichy’s result holds, as acorollary, for any Kripkean system.In contrast, we can easily check that de re and de dicto modalities entail each other in Tm ∗ .This is a direct consequence of (NBF) , (PBF) and the definition of ∃ and ♦ . However, it couldbe possible to separate these notions if we consider the first definition of the universal quantifierwe propose here, as being a (non-deterministic) Tm ∗ -conjunction, namely the multioperator˜ ∀ , instead of the deterministic operator ˜ ∀ d adopted for Tm ∗ (recall Subsection 2.1). Indeed,consider an unary predicate symbol P and let A be a first-order structure for Tm ∗ such that ∅ 6 = a A ( P ) ⊂ c A ( P ) = U . Then, for every v and s , { v ( P ( x ) , s ′ ) : s ′ ∈ E x ( s ) } = { C − , C + } .This produces the following scenario: X ˜ ♦ ( X ) ˜ ∀ ( ˜ ♦ ( X )) ˜ ∀ ( X ) ˜ ♦ (˜ ∀ ( X )) { C − , C + } { T + , C + } { C + } { F − , C − } V Therefore, by taking v ( ♦ ∀ xP x, s ) = C − , the formula ∀ x ♦ P x → ♦ ∀ xP x will be refuted. Thatis, (NBF) will be invalidated. Of course this is equivalent to invalidate the consequence relation (cid:3) ∃ xP x | = ∃ x (cid:3) P x . Indeed, in the same situation as above, by considering the first defini-tion of the existential quantifier as being a (non-deterministic) Tm ∗ -disjunction, namely themultioperator ˜ ∃ instead of ˜ ∃ d , we will have: See [26, p. 622–633].
Philosophical questions concerning first-order modal logic X ˜ ∃ ( X ) ˜ (cid:3) (˜ ∃ ( X )) ˜ (cid:3) ( X ) ˜ ∃ ( ˜ (cid:3) ( X )) { C − , C + } { T + , C + } V { F − , C − } { C − } and so, by taking v ( (cid:3) ∃ xP x, s ) = C + the modal inference (cid:3) ∃ xP x | = ∃ x (cid:3) P x will be invalidated.Observe that, in order to axiomatize ˜ ∀ in Tm ∗ , it is enough to delete axiom (NBF) from theaxiomatization of Tm ∗ given in Definition 2.2. It is also worth noting that the non-deterministicquantifier ˜ ∀ (as well as its dual ˜ ∃ ) still satisfies the Barcan formulas (BF) and (CBF) (ofcourse, in the case of the existential quantifier, both formulas must be expressed in terms of ♦ ). Thus, the use of this quantifier just blocks (NBF) , while preserving both Barcan formulas.In particular, (PBF) is validated by the non-deterministic quantifier ˜ ∀ . Indeed, it is easy tocheck that the situation concerning (PBF) is as follows: X ˜ ♦ (˜ ∀ ( X )) ˜ ∀ ( ˜ ♦ ( X )) F − / ∈ X V or { T + , C + } { C + } F − ∈ X { C − , F − } { F − } The definition of an intuitive notion of universal quantifier in Tm ∗ which also rejects (PBF) deserves future research.Some philosophers may defend that the distinction between de re and de dicto modalitiesis very important from a metaphysical point of view. Others, like Abelardo, could claim thatthere is no such distinction. The point here is that Kripke’s relational semantics is not neutralwith respect to that polemics: it forces us to go against Abelard, that is, it forces us to admita metaphysical distinction that, according to Quine, throws us into the jungle of Aristotelianessentialism . In contrast, Tm ∗ offers a formal interpretation of “necessary” such that, inQuinean terms, is much more civilized. Finally, a famous problem of quantified modal logic will be briefly discussed here from theperspective of the non-deterministic semantics: the so-called
Barcan formulas (BF) and (CBF) ,which were already mentioned above.In standard Kripke semantics with constant domains, the formula (BF) is valid in thosemodels in which the relational accessibility between possible world is, at least, reflexive andsymmetrical, while (CBF) is not valid in general. On the other hand, there are variabledomains in which neither (BF) nor (CBF) are valid. Thus, Kripke models offer an apparatussufficiently refined to invalidate (CBF) and, in some cases, even (BF) .As analyzed above, both laws (BF) and (CBF) hold in any model of Tm ∗ — even whenconsidering non-deterministic quantifers in order to block (NBF) . From this, the reader couldask if Ivlev’s semantics is less expressive than Kripke models with variable domains, as long as See [33]. It seems that Tm ∗ semantics is immune to Quine’s three criticisms to modal logic. The first ofthese criticisms is that, in the propositional context, being necessary would be equivalent to being tautological.But this is not the case in Ivlev semantics, for a very simple reason: we can prove that α ∨ ¬ α is always true,but if the value of α is C + , we could choose a false value for (cid:3) ( α ∨ ¬ α ). The second criticism concerns, roughlyspeaking, to the fact that identities are always necessary. As we have seen, this is not necessarily the case inour approach. The third criticism concerns the modal commitment of a kind of essentialism that, as arguedabove, does not apply to our systems. See [20, p.244-249]. In varying domain models, (BF) is valid if and only if they are anti-monotonic, that is, if a world w isrelated to w ′ , then the domain of w ′ is a subset of the domain of w . In contrast, (CBF) is valid if and only themodels are monotonic, that is, if w is related to w ′ , then the domain of w is a subset of the domains of w ′ . See[17, p. 108-112]. Philosophical questions concerning first-order modal logic the former collapses formulas that are distinguishable in Kripke’s approach. It will be arguedthat this kind of conclusion is, at the very least, hasty, and that the situation is a little morecomplex than it seems.Given a wff α , consider the following notation: (cid:3) n α ≡ def (cid:3)(cid:3) . . . (cid:3) | {z } n times α Now, let us consider now the following generalized Barcan and generalized converse Barcan formulas: (BF n ) ∀ x (cid:3) n α → (cid:3) n ∀ xα (CBF n ) (cid:3) n ∀ xα → ∀ x (cid:3) n α In any modal logic that validates the necessitation rule and axiom (K) , if (BF) holds, then (BF n ) will hold too. Those considerations also apply to (CBF) and (CBF n ) .That is not the case in Ivlev’s semantics. Consider, for instance, a signature with an unarypredicate symbol P . Let A be a structure for Tm ∗ with domain U such that a A ( P ) = U and c A ( P ) = ∅ . Observe that v ( P x, s ) = T + (hence v ( ∀ xP x, s ) = T + ) for every v and s . Now, let v be a valuation over A such that v ( (cid:3) P x, s ) = T + and v ( (cid:3) ∀ xP x, s ) = C + , for any s . From this, { v ( (cid:3)(cid:3) P x, s ′ ) : s ′ ∈ E x ( s ) } is a set of designated values, for any s . Hence v ( ∀ x (cid:3)(cid:3) P x, s ) isdesignated, for any s . On the other hand, v ( (cid:3)(cid:3) ∀ xP x, s ) is not designated, for every s . Then,( A , v ) does not validate (BF ) . This argument can be easily generalized to any (BF n ) for n > v ′ over the structure A defined above such that, for any s , v ( (cid:3) P x, s ) = C + and v ( (cid:3) ∀ xP x, s ) = T + . Thus, { v ( (cid:3)(cid:3) P x, s ′ ) : s ′ ∈ E x ( s ) } is a set ofnon-designated values, for any s , and so v ( ∀ x (cid:3)(cid:3) P x, s ) is not designated, for any s . On theother hand, v ( (cid:3)(cid:3) ∀ xP x, s ) is designated, for every s . This shows that ( A , v ) does not validate (CBF ) and, again, the argument can be easily generalized to any (CBF n ) for n > ∀ x (cid:3) P x can be interpreted as saying thatall the objects of the domain have essentially the propriety P . As we saw in Subsection 3.3,this is a case of the de re modality. The formula (cid:3) ∀ xP x , in turn, is a de dicto modality. An essentialist does not want the collapse of modalities de re and de dicto . That is whythe non-deterministic semantics proposed here could not be interesting for this philosophicalperspective. In contrast, this Nmatrix semantics separates levels of essentialism: someone couldhold that there is no first-level essentialism, while accepting the second-level one: the fact thatall object have a property as essentially essential does not imply that it is necessarily necessarythat all the objects have this property. This new philosophical perspective could be called a multi-leveled essentialism .It is worth noting that, for Kripke, the relation between worlds must be, at least, reflexive. In this case, axiom (T) holds, hence the collapse of (BF) and (BF n ) is unavoidable, and thesame holds for (CBF) and (CBF n ) . In turn, the system Tm ∗ is an interesting case in which,even with axiom (T) holding, there is no collapse at all, as we have seen above.Finally, let us consider the strict implication , which is defined as usual as In this sense, essentialism is “the doctrine that (at least some) objects have (at least some) essentialproperties. This characterization is not universally accepted, but no characterization is; and at least this one hasthe virtue of being simple and straightforward” (see [38]). According to the author, the modal characterizationof the essencialism is “ P is an essential property of an object o just in case it is necessary that o has P , whereas P is an accidental property of an object o just in case o has P but it is possible that o lacks P ”. Formally, if c is an individual constant representing o , we symbolize “ P is an essential propriety of o ” by (cid:3) P c , whereas “ P is an accidental property of P ” is symbolized by P c ∧ ♦ ¬ P c . Segerberg defends that, against “followers of Kripke’s terminology”, normal modal logics should cover thosemodels whose accessibility relation is empty, see [40, p. 12].
Considering other Ivlev-like modal systems α J β ≡ def (cid:3) ( α → β )Let us consider the following version of (BF) and (CBF) : (BF J ) ∀ x (cid:3) α J (cid:3) ∀ xα (CBF J ) (cid:3) ∀ xα J ∀ x (cid:3) α In Kripke semantics, (BF) implies (BF J ) , by the necessitation rule. But that is not the casewith Nmatrix semantic: just take an instance of (BF) as contingently true in some ( A , v ). The argument is analogous with respect to (CBF) and (CBF J ) .Because of this, it is not possible to support an essentialist view with respect to strictimplication but not with respect to material implication in the usual Kripkean context. Wecould call the stricter rules (BF J ) and (CBF J ) as strict essentialism . In this sense, the non-deterministic semantics of Tm ∗ can model a version of essentialism, namely (BF) and (CBF) ,which is weaker than strict essentialism.For these reasons, we believe that the non-deterministic semantics for first-order modallogics proposed here can uncover subtleties that were previously invisible to us. This is anatural consequence of the fact that our eyes are too accustomed to facing these problemsthrough the glasses of the possible worlds semantics. The previous sections were devoted to analyze first-order extensions of the four-valued Ivlev sys-tem Tm . In previous papers ([10, 11, 12]) we investigated other Ivlev-like modal propositionalsystems, characterized by four-valued, six-valued and eight-valued Nmatrices.Concerning four-valued systems, the extension to first-order languages of some of them,such as T4m and
T45m , were mentioned in Subsection 3.2. The soundness and completenessresults, as well as all topics discussed above about Tm ∗ , can be easily adapted to the systems T4m ∗ and T45m ∗ .Recall from Section 1 that the four-valued non-deterministic semantics for Tm can bedescribed in terms of the concepts of actually true and contingently true . From this, thepredicate symbols in a structure A for Tm ∗ are interpreted in terms of the mappings a A and c A (which, as observed in Remark 2.6, is equivalent to consider a function P A : U n → V , for every n -ary predicate symbol P ). However, as discussed in Part I of this paper (see [12, Section 1]),this is a particular case of a more general situation involving eight truth-values, each of oneexplained in terms of the modal concepts of necessarily true , possibly true and actually true : T + : necessarily, possibly and actually true; C + : contingently and actually true; F + : impossible, possibly false but actually true; I + : necessary true, impossible and actually true; T − : necessarily and possibly true but actually false; C − : contingently and actually false; In fact, those are the original formulas considered by Ruth C. Barcan in [4]. That is, consider a pair ( A , v ) and an instance γ of (BF) such that, for every s , v ( γ, s ) = C + . Then, (cid:3) γ isan instance of (BF J ) such that v ( (cid:3) γ, s ) is not designated, for evey s . Considering other Ivlev-like modal systems F − : impossible, possibly false and actually false; I − : necessarily true, impossible and actually false.Under this perspective, + = { T + , C + , F + , I + } represents being actually true (the designatedtruth-values), while being actually false is given by − = { T − , C − , F − , I − } (the undesignatedtruth-values). This produces an eight-valued non-normal version of K that we called Km .However, as observed in Part I, the values I + and I − represent very artificial situations. Thislead us to consider six-valued structures, apt to deal with a deontic non-normal logic called Dm , as well as some other axiomatic extensions of it. The interpretation of a predicate symbol P in eight-valued structures for Km and in six-valued structures for Dm , respectively, arerepresented as follows: a A n A p A T + T − C + I − I + F + C − F − a A n A p A F + C + T + T + T − C − F − Recall from [12, Section 4] that the multioperations for the Nmatrix characterizing the logic Km are as follows, where ♦ α := ¬ (cid:3) ¬ α , α ∨ β := ¬ α → β and α ∧ β := ¬ ( α → ¬ β ) and + isthe set of designated values: ˜ ¬ ˜ (cid:3) ˜ ♦ T + { F − } + + C + { C − } − + F + { T − } − − I + { I − } + − T − { F + } + + C − { C + } − + F − { T + } − − I − { I + } + − ˜ → T + C + F + I + T − C − F − I − T + { T + } { C + } { F + } { I + } { T − } { C − } { F − } { I − } C + { T + } { T + , C + } { C + } { I + } { T − } { T − , C − } { C − } { I − } F + { T + } { T + } { T + } { I + } { T − } { T − } { T − } { I − } I + { I + } { I + } { I + } { I + } { I − } { I − } { I − } { I − } T − { T + } { C + } { F + } { I + } { T + } { C + } { F + } { I + } C − { T + } { T + , C + } { C + } { I + } { T + } { T + , C + } { C + } { I + } F − { T + } { T + } { T + } { I + } { T + } { T + } { T + } { I + } I − { I + } { I + } { I + } { I + } { I + } { I + } { I + } { I + } Considering other Ivlev-like modal systems ˜ ∨ T + C + F + I + T − C − F − I − T + { T + } { T + } { T + } { I + } { T + } { T + } { T + } { I + } C + { T + } { T + , C + } { C + } { I + } { T + } { T + , C + } { C + } { I + } F + { T + } { C + } { F + } { I + } { T + } { C + } { F + } { I + } I + { I + } { I + } { I + } { I + } { I + } { I + } { I + } { I + } T − { T + } { T + } { T + } { I + } { T − } { T − } { T − } { I − } C − { T + } { T + , C + } { C + } { I + } { T − } { T − , C − } { C − } { I − } F − { T + } { C + } { F + } { I + } { T − } { C − } { F − } { I − } I − { I + } { I + } { I + } { I + } { I − } { I − } { I − } { I − } ˜ ∧ T + C + F + I + T − C − F − I − T + { T + } { C + } { F + } { I + } { T − } { C − } { F − } { I − } C + { C + } { F + , C + } { F + } { I + } { C − } { F − , C − } { F − } { I − } F + { F + } { F + } { F + } { I + } { F − } { F − } { F − } { I − } I + { I + } { I + } { I + } { I + } { I − } { I − } { I − } { I − } T − { T − } { C − } { F − } { I − } { T − } { C − } { F − } { I − } C − { C − } { F − , C − } { F − } { I − } { C − } { F − , C − } { F − } { I − } F − { F − } { F − } { F − } { I − } { F − } { F − } { F − } { I − } I − { I − } { I − } { I − } { I − } { I − } { I − } { I − } { I − } The Nmatrix for Dm is obtained from this by eliminating the values I + and I − , while theNmatrix for Tm is obtained by additionally removing the values F + and T − . Indeed, andas shown in [12, Proposition 5.5], A Tm ⊆ sm A Dm ⊆ sm A Km . Here, A Dm and A Km denoterespectively the multialgebras for Dm and Km , and ⊆ sm is the ‘submultialgebra’ relation. From an axiomatic point of view, Dm is obtained from Tm by replacing (T) by an attenuatedversion of this axiom, namely, the well-known deontic axiom: (D) (cid:3) α → ♦ α From this, first-order extensions Dm ∗ and Km ∗ of Dm and Km can be defined, in the sameway as Tm ∗ was obtained. Indeed, the definition of the corresponding Hilbert calculi is obvious:it is enough to add to the Hilbert calculi introduced in Part I for these logics the axioms and theinference rule for the universal quantifier considered for Tm ∗ . With respect to semantics, let V k be the set of k truth-values considered above, for k = 6 ,
8. Then, it is enough to expand thecorresponding multialgebras with suitable multioperators ˜ Q k : ( P ( V k ) − {∅} ) → ( P ( V k ) − {∅} )for every quantifier Q ∈ {∀ , ∃} and k = 6 , V k defined in Subsection 2.1. From the eight-valued multifuncions ˜ ∧ and˜ ∨ for conjunction and disjunction displayed above it is straightforward to define, respectively,˜ ∀ k and ˜ ∃ k for k = 6 , (NBF) , the deterministicversion ˜ Q dk of these quantifiers can be easily defined by removing the occurrences of F − or T + from ˜ Q k ( X ) in the instances of X where ˜ Q k ( X ) is not a singleton. Namely, for k = 6 , ∀ dk ( X ) := { C − } for X = { C + , C − } or X = { C + , C − , T + } , while ˜ ∃ dk ( Y ) := { C + } for Y = { C + , C − } or Y = { C + , C − , F − } . We can extend this approach by considering the systems D4m ∗ , D45m ∗ , K4m ∗ and K45m ∗ obtained from the corresponding propositional systemsstudied in Part I. Then, soundness and completeness of these systems with respect to the By comparing the figure for a A ( P ) and c A ( P ) displayed in Section 1 with the ones for a A ( P ), n A ( P ) and p A ( P ) displayed above it is easy to see that, when deleting the additional values, n A ( P ) = a A ( P ) \ c A ( P ) and p A ( P ) = a A ( P ) ∪ c A ( P ) for every predicate symbol P . Final Remarks first-order Nmatrix semantics is easily obtained by adapting and extending the correspondingproofs for Tm ∗ detailed in the previous sections. Of course every n -ary predicate symbol P should be interpreted in a six-valued or eight-valued first-order structure A as a triple P A :=( a A ( P ) , n A ( P ) , p A ( P )) of subsets of U n . Once again, the details of these constructions are leftto the interested reader.None of the above mentioned six-valued and eight-valued systems validate axiom (T) . In-validating (T) is not just a merely formal curiosity. We know that (BF) and (BF J ) collapsein reflexive Kripkean models. In Tm ∗ , (BF J ) implies (BF) and that seems to be a naturalproperty, since strict implication is stronger than material implication. But that collapse is adirect consequence of axiom (T) . The same consideration can be done with respect to (CBF) and (CBF J ) . In any Ivlev-like first-order system in which (T) does not hold, the formulas (BF) and (BF J ) will be totally independent, and the same holds for (CBF) and (CBF J ) . In this article, our investigations on non-normal modal logics with finite-valued non-deterministicmatrix semantics developed in Part I (see [12]) were extended to first-order languages. By sim-plicity, and in order to fix the main ideas and the many advantages that this proposal can offerto the subject of first-order modal logics, just the quantified version of system Tm , called Tm ∗ ,was analyzed in detail. The extension of this approach to other modal systems based on T4m , T45m , Dm , D4m , D45m , Km , K4m or K45m should not be too difficult, as outlined inSection 4.Concerning the formulas (BF) and (CBF) , we have proposed here a semantics in whichthese two formulas, at least at the first level, collapse (see Subsection 3.4). But this is not aconsequence of the semantic clauses of atomic and propositional formulas. In fact, if our clausesregarding the universal operator were different, we could invalidate any of these formulas, as inthe case of axiom (NBF) (recall Subsection 3.3). As mentioned there, it should be interestingconsidering another notion of universal quantifier in which (PBF) would be blocked as well.It worth noting that the way adopted in this paper defining the deterministic versions ofthe quantifiers intends to be the closest formal counterpart to our language intuitions involvingterms like “necessarily” and “for all”. This is not to say that we cannot discover another formalversions as much (or even more) intuitive than those presented here. We have not ruled outthis possibility yet.It is well-known that first-order classical logic is not decidable. But certain fragments of thislogic are, in turn, decidable. An interesting case is its monadic fragment, that is, the fragmentin which every predicate is unary and no function symbols are allowed. By contrast, Kripke hasshown that almost all monadic first-order modal systems are undecidable. But this result doesnot automatically apply to Tm ∗ or any monadic fragment of the systems treated here. In fact,all propositional Ivlev-like systems proposed in Part I are decidable. It would be interesting toknow how many of their monadic extensions are decidable.Even though Tm ∗ is a system that does not compromise with the Kripkean thesis thatproper names are rigid designators while definite descriptions are non-rigid designators, it wouldbe interesting to consider a four-valued non-deterministic modal semantics in which this occurs. As a matter of fact, the observation made in Remark 2.6 for Tm ∗ can be applied to these six-valuedand eight-valued modal systems. Namely, interpreting a n -ary predicate symbol P by means of a triple P A =( a A ( P ) , n A ( P ) , p A ( P )) is equivalent to consider functions P A : U n → V k for k = 6 ,
8. The latter approach isanalogous to the non-deterministic first-order structures for paraconsistent logics based on Nmatrices consideredin [13]. See [27].
Final Remarks A possible way would be extending the interpretation mappings a and c to function symbols,while the individual constants would receive a unique interpretation in a given structure A .Besides, individual variables would receive two kind of interpretations by means of the as-signments, the actual one and the contingent one. We are aware, however, of the technicaldifficulties of this possible solution.First-order logic has difficulties for dealing with non-existent object that has a name, like“Pegasus”. That happens because to each constant in the language we associate an individualin the domain. Since the existential quantifier traverses individuals in the domain of structure,in classical logic being named is equivalent to existing. Thus, if Pegasus is a winged horse, thenPegasus is a name of an individual of the domain and, thus, Pegasus exists. But we know thatPegasus is a winged horse that does not exist. In order to block this inference, we should use theso called Free Logics. It is a relevant philosophical problem to decide whether Pegasus is a beingthat does not exist but possibly exists. Free Logics have made a very fruitful contribution tothese investigations, especially when combined with semantics of possible worlds. We have noreason to be skeptical of the success of addressing these problems combining non-deterministicsemantics with Free Logics.
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